Cliquez pour un

discussion de certaines omissions.

S'il vous plaît écrivez-moi si

vous pensez qu'il y a une grosse erreur dans

mon classement (ou une erreur dans certaines biographies).

De toute évidence, les rangs relatifs de Fibonacci et

Ramanujan, ne satisfera jamais toutes les raisons secondaires

pour leur "grandeur" est différent.

Je suis sûr d'avoir oublié de grands mathématiciens

qui bien sûr appartient à cette liste.

S'il vous plaît email et dites-moi!

Voici les meilleurs mathématiciens de

ordre chronologique (année de naissance).

(En passant, le classement attribué à un mathématicien apparaîtra si vous

placez le curseur en haut du nom en haut de sa mini-bio.):

Il est peu connu sur les premières mathématiques, mais le célèbre

Ishango Bonedu début de l'âge de pierre en Afrique a des caractères de timbre qui indiquent

arithmétique. Les balises contiennent six nombres premiers (5, 7, 11, 13, 17, 19)

dans l’ordre, bien que ce soit probablement une coïncidence.Les artefacts avancés de l'ancien royaume d'Égypte

et la civilisation Indus-Harrapa

implique de fortes compétences en mathématiques, mais le premier

preuve écrite de dates arithmétiques avancées de Sumeria,

où les tablettes d'argile vieilles de 4500 ans montrent la multiplication et

problèmes de division; Le premier boulier peut être à propos de cet ancien.

Il y a 3600 ans, les comprimés mésopotamiens

tables de carrés, cubes, inverses et même logarithmes

et fonctions trigonométriques,

utilise un système de valeur de position primitive (en base 60 et non 10).

Les Babyloniens étaient familiers avec le théorème de Pythagore,

solutions aux équations carrées,

même des équations cubiques (bien qu’elles n’aient pas de solution générale),

et enfin développé des méthodes pour estimer

termes d'intérêts composés.

Les Grecs ont emprunté aux mathématiques babyloniennes, qui étaient les plus

avancé par certains avant les Grecs; mais c'est pas

ancien mathématicien babylonien dont le nom est connu.Il y a aussi au moins 3600 ans, l'écrivain égyptien Ahmes

produit un manuscrit connu

(maintenant appeléPapyrus Rhind)

elle-même une copie d'un texte de la fin du Moyen Empire.

Il a montré des méthodes simples d’algèbre et

y compris une table qui fournit optimale

expression au moyen de factions égyptiennes.

(Aujourd’hui, les factions égyptiennes conduisent à remettre en question la théorie des nombres

problèmes sans applications pratiques, mais ils peuvent avoir

avait une valeur pratique pour les Egyptiens.

Partageant 17 autobus à grains entre 21 travailleurs, il

l'équation 17/21 = 1/2 + 1/6 + 1/7 a une valeur pratique,

surtout par rapport à

la décomposition "gourmande"

17/21 = 1/2 + 1/4 + 1/17 + 1/1428.)Les pyramides démontrent que les Egyptiens

était adepte de la géométrie, bien que peu de preuves écrites subsistent.

Babylone était bien plus avancée que l’Égypte en arithmétique et en algèbre;

Cela était probablement dû, au moins en partie, à leur système de valeur de position.

Mais même si leur système base 60 survit (par exemple, dans la division)

en heures et degrés en minutes et secondes) le Babylonien

notation, qui utilisait l'équivalent de IIIIII XXXXIII

dénotant 417 + 43/60 était ingérable par rapport à

"dix chiffres en hindous."

(En 2016, les historiens ont été surpris de décoder les anciens

textes et trouver des calculs astronomiques très sophistiqués

du chemin de Jupiter.)Les Egyptiens ont utilisé l'approche

π ≈ (4/3)^{4}(dérivé de l'idée qu'un

Le cercle de diamètre 9 a à peu près la même surface qu’un carré de la page 8).

Bien que l'ancien mathématicien hindou Apastambha ait réalisé

une bonne approche de√2et les anciens Babyloniens

toujours mieux√2, aucun de ces anciens

les cultures ont atteint unπapproche aussi bonne que celle de l'Egypte,

ou mieux queπ ≈ 25/8, à l'ère d'Alexandrie.La floraison soudaine des mathématiques en Inde et

La Grèce doit beaucoup aux anciennes mathématiques de l’Égypte et de Babylone.

**Premiers mathématiciens védiques**

Les plus grandes mathématiques avant

L'âge d'or en Grèce était aux Indes

début de la civilisation védique (hindoue).

Vedics a compris la relation entre la géométrie

et arithmétique, astronomie développée, astrologie, calendriers,

et utilisé des formes mathématiques dans certains rituels religieux.Le premier mathématicien à qui des doctrines particulières peuvent être attribuées

était Lagadha, qui a apparemment vécu environ 1300 av. et

a utilisé la géométrie et la trigonométrie élémentale pour son astronomie.

Baudhayana vécut environ 800 ans av. et a également écrit sur l'algèbre et la géométrie;

Yajnavalkya a vécu à peu près à la même époque et est considéré comme le meilleur

approche deπ.

Apastambha a effectué le travail résumé ci-dessous;

d'autres premiers mathématiciens védiques résolurent des équations carrées et simultanées.D'autres cultures anciennes ont également développé les mathématiques.

Les anciens Mayas avaient apparemment un système de valeur de position

avec zéro avant les Hindous ont fait;

L'architecture aztèque implique des compétences pratiques en géométrie.

La Chine ancienne a probablement développé les mathématiques,

En fait, la première preuve connue du taux de Pythagore

trouvé dans un livre chinois (Zhoubi Suanjing)

quipeut-êtrea été écrit vers 1000 av.

(environ 624 – 546 av. J.-C.) domaine grec

Thalès était le chef des "Sept femmes" dans la Grèce antique,

et a été appelé "le père de la science"

"Le fondateur de la géométrie abstraite"

et "le premier philosophe".

Thales aurait étudié les mathématiques

sous les Egyptiens, qui à leur tour étaient au courant des personnes beaucoup plus âgées

mathématiques de la mésopotamie.

Thales a peut-être inventé le terme compas et règle

construction.

Plusieurs théories de base sur les triangles

attribué à Thales, y compris la loi des triangles similaires

(que Thalès utilisait pour calculer la hauteur de la grande pyramide)

et "Théorème de Thales" lui-même: le fait que tout angle inscrit dans

un demi-cercle est un angle droit.

(Les autres "théorèmes" ressemblaient probablement plus à des axiomes connus,

mais Thales a prouvé que le théorème de Thales utilisait deux des

ses autres théorèmes On dit que Thalès a ensuite sacrifié

un taureau pour célébrer ce qui aurait pu être le premier

preuves mathématiques en Grèce.)

Thales l'a noté, vu un segment de ligne

de longueurx, un segment de longueurx / kpeut être construit

en construisant d'abord un segment de longueurkx.Thalès était aussi un astronome; il a inventé le calendrier de 365 jours,

introduit l'utilisation des mines d'ursa pour trouver le nord,

a inventé la projection cartographique gnomonique (la première de nombreuses

méthodes connues aujourd'hui pour cartographier (une partie de) la surface d'une sphère

pour un avion,

et est la première personne supposée avoir correctement prédit une éclipse solaire.

Ses théories de la physique sembleraient charmantes aujourd’hui, mais il

semble avoir été le premier à décrire le magnétisme et l'électricité statique.

Aristote a dit: "Pour Thales était la première question

pas ce que nous savons, mais comment le savons-nous? "

Thales était également un politicien, un stratège éthique et militaire.

On dit qu'il a loué tous les presses à olives disponibles

après avoir demandé une bonne saison des olives; il n'a pas

pour la richesse elle-même, mais comme démonstration de son utilisation

de l'intelligence dans l'entreprise.

Les écrits de Thales n'ont pas survécu et ne sont connus que pour être utilisés.

Depuis ses célèbres théories de la géométrie étaient probablement déjà connues

dans l'ancienne Babylone, sa signification découle de donner

notions de preuve mathématique et de méthode scientifique

aux vieux Grecs.

Alors que plusieurs vieux mathématiciens étaient préoccupés par le côté pratique

Les calculs, les mathématiques modernes ont commencé avec le poids grec

sur les preuves et la philosophie. Pythagore et Parménide d’Élée aussi

a joué un rôle clé dans le développement de Thales.

Ces idées ont conduit aux écoles de Platon, Aristote et Euclide,

et une floraison intellectuelle incomparable

Renaissance de l'Europe.L’élève et successeur de Thales était Anaximander, souvent appelé

"Premier chercheur" au lieu de Thalès: ses théories étaient plus

fermement basé sur l'expérimentation et la logique, tandis que Thales poursuit

fier de certaines interprétations animistes.

Anaximander est connu pour l'astronomie, la cartographie et les cellules solaires,

et a également expliqué une théorie de l'évolution, que les espèces

en quelque sorte développé à partir de poisson primordial!

L'étudiant le plus célèbre d'Anaximandre fut à son tour Pythagore.

(environ 630-560 av. J.-C.) Inde

ils

Dharma Sutracomposé par Apastambha contient mensuration

techniques, de nouvelles techniques d'ingénierie géométrique, une méthode de

algèbre, et ce qui peut être une preuve précoce du théorème de Pythagore.

Apastambha utilise l'excellent (fraction continue)

approche√2, un résultat qui est probablement dérivé de

7 577/408

un argument géométrique.Apastambha s’appuie sur les travaux d’anciens érudits védiques,

surtout Baudhayana aussi

comme les mathématiciens Harappa et (probablement) mésopotamiens.

Sa notation et ses preuves étaient primitives, et c'est petit

certitude sur sa vie.

Cependant, des commentaires similaires s’appliquent à Thales of Miletus, donc ça marche

juste pour mentionner Apastambha (qui était peut-être ce

mathématicien védique le plus créatif avant Panini)

avec Thales comme l’un des

premiers mathématiciens dont les noms sont connus.

(vers 578-505 av. J.-C.) domaine grec

Pythagore, parfois appelé "le premier philosophe"

étudié sous Anaximandre, Egyptiens, Babyloniens,

et mystérieux Pherekydes (de qui

Pythagore a acheté une croyance en la réincarnation); il est resté

le plus influent des premiers mathématiciens grecs.

Il est crédité pour avoir été le premier à utiliser des axiomes

et des preuves déductives, son influence sur Platon et Euclide peut donc être énorme;

Il est généralement crédité de beaucoup de

Livres I et II d’Euclide séléments.

Lui et ses étudiants ("pythagoriciens") étaient des mystiques ascétiques

pour qui les mathématiques étaient en partie un outil spirituel.

(Certains occultistes considèrent Pythagore comme un sorcier et fondateur de mystiques

philosophe).

Pythagore était très intéressé par l'astronomie et semble avoir été

premier homme à se rendre compte que la terre était un monde semblable aux autres planètes.

Lui et ses successeurs ont commencé à étudier la question des mouvements planétaires,

Cela ne serait pas résolu pendant plus de deux millénaires.

les motsphilosophieetmathématiquesdit avoir

été inventé par Pythagore.

Il doit avoir inventéCoupe de Pythagoreet intelligent

bouchée de vin qui punit une boisson gourmande qui remplit sa coupe au sommet

en utilisant la pression du siphon pour vider la tasse.Malgré l'importance historique de Pythagor, je l'ai peut-être trop classé:

de nombreux résultats des pythagoriciens étaient

à cause de leurs étudiants; aucun de leurs écrits ne survit; et quoi

est connu pour être utilisé, et peut-être

exagéré par Platon et d’autres.

Certaines idées qui lui sont attribuées ont probablement été éclairées au préalable par les successeurs

comme Parménide d'Elée (environ 515-440 av. J.-C.).

Les archéologues estiment maintenant qu'il n'a pas été le premier à inventer le diatonique

échelle:

Voici une chanson diatonique d'Ugarit

qui se déroule à Pythagore avec huit siècles.Parmi les étudiants de Pythagore figuraient Hippasus de Metapontum,

le célèbre anatomiste et médecin Alcmaeon (qui

était le premier à dire que penser se passait dans le cerveau au lieu du coeur)

Milo ou Croton,

et la fille de Milo, Theano (qui peut-être était la femme de Pythagore).

le termePythagorea également été adopté par de nombreux disciples qui ont vécu plus tard;

Parmi ces disciples figurent le filole de Croton,

le philosophe naturel Empedocles et plusieurs autres Grecs célèbres.

Le successeur de Pythagore était apparemment Theano lui-même:

Les pythagoriciens étaient l’une des rares vieilles écoles

pratiquer l'égalité.Pythagore a découvert que les intervalles harmoniques dans la musique sont basés

nombres rationnels simples.

Cela a conduit à une fascination pour les nombres entiers

et la numérologie mystique;

il est parfois appelé "le père du nombre"

et une fois dit "Le nombre compte l'univers."

(Sur les bases mathématiques pour la musique, écrivait plus tard Leibniz,

"La musique est la joie que l'âme humaine éprouve en comptant

Sans être conscient que ça compte. "

Autres mathématiciens ayant étudié l'arithmétique

de musique comprenant Huygens, Euler et Simon Stevin.)

Quelques chiffresunetbLes pythagoriciens en étaient conscients

des trois manières différentes:

(A + b) / 2(moyenne arithmétique),

√ (ab)(moyenne géométrique), et

2AB / (a + b)(agent harmonique).La phrase pythagore était connue bien avant Pythagore,

mais il a souvent été crédité (avant la découverte d'un ancien texte chinois)

avec le premierpreuve.

Il a peut-être découvert la forme paramétrique simple

des triplets pythagoriciens primitifs(xx-yy, 2xy, xx + yy),

Bien que la première mention explicite de ceci puisse être dans le commentaire d'Euclideéléments.

Parmi les autres découvertes de l’école de Pythagore:

construction du pentagone habituel,

notions de nombres parfaits et amicaux,

nombres polygonaux,

relation en or (attribuée à Theano),

trois des cinq solides solides (attribués à Pythagore eux-mêmes),

et nombres irrationnels (attribués à Hippasus).

On dit que les résultats

des nombres irrationnels, les pythagores ont tellement bouleversé

ils ont jeté Hippase dans la mer!

(Une autre version bannie par Hippasus pour divulgation

le secret de la construction de la sphère qui entoure

et dodécaèdre.)Outre Parménide, les célèbres successeurs de Thalès et de Pythagore

inclure Zeno of Elea (voir ci-dessous),

Hippocrate de Chios (voir ci-dessous),

Platon d'Athènes (environ 428 à 348 av. J.-C.),

Théétète (voir ci-dessous) et Archytas (voir ci-dessous).

Ces premiers Grecs sont entrés dans un âge d'or des mathématiques

et philosophie

inégalée en Europe pour la Renaissance.

L'accent était mis sur les mathématiques pures plutôt que pratiques.

Platon (qui se trouve n ° 40 sur la célèbre liste de Michael Hart de

les personnes les plus influentes de l'histoire)

décidé que ses érudits devraient faire

construction géométrique uniquement avec compas et règle

plutôt qu'avec "les outils du charpentier" comme

les dirigeants et les rangs.

(environ 520-460 av. J.-C.) Gandhara (Inde)

La grande réussite de Panini a été son étude

de la langue sanskrit, en particulier dans son texteAshtadhyayi.

Bien que ce travail puisse être considéré comme la toute première étude

de linguistique ou de grammaire, il a utilisé une élégance non évidente

Cela ne serait pas assimilé en Occident avant le 20ème siècle.

La linguistique peut sembler une qualification improbable pour un "grand mathématicien"

Mais la théorie des langues est un domaine des mathématiques.

Les travaux d'éminents linguistes et informaticiens du XXe siècle

comme Chomsky, Backus, Post et Church

ressemble au travail de Panini 25 siècles plus tôt.

L'étude systématique du sanscrit par Panini a peut-être inspiré

le développement de la science et de l'algèbre indiennes.

Panini a été appelé "Indian Euclid" depuis sévérité

de sa grammaire est comparable à la géométrie d'Euclide.Bien que ses beaux textes aient été préservés, il n’ya guère d’autre

est connu à propos de Panini. Certains érudits placeraient leurs dates un siècle

plus tard que montré ici; il peut être ou ne pas être la même personne

comme le célèbre poète Panini.

En tout cas, il était la dernière école sanskrite védique

Définition: son texte forme la transition vers

Période sanskrit classique.

Panini a été appelé "l'un des plus innovants

les gens tout au long du développement des connaissances; "

sa grammaire "un des plus grands monuments de l'intelligence humaine".

(environ 495-435 av. J.-C.) domaine grec

Zénon, élève de Parménide, avait une grande renommée

la Grèce antique.

Cette renommée, qui continue à ce jour, est en grande partie due à

ses paradoxes de l'infini, par exemple son argument

qu'Achille ne peut jamais attraper la tortue (quand Achille vient

sur la dernière position de la tortue la tortue est allée).

Bien que certains considèrent ces paradoxes comme de simples erreurs,

Ils font de l'embonpoint depuis plusieurs siècles.

C’est à cause de ces paradoxes que

utilisation d'infinitésimaux, qui constituent la base de l'analyse mathématique,

a été considéré comme un héduriste non strict et

est finalement considéré comme sain qu'après le grand travail

Strigoristes du 19ème siècle, Dedekind et Weierstrass.

Le paradoxe de la flèche de Zénon (à la fois, une flèche est fixée

d'où vient son mouvement?) a emprunté un nom pour

L'effet Zeno quantique, un paradoxe de la physique quantique.Eubulides ou Milet

était un autre grec ancien connu pour les paradoxes,

par exemple "Cette déclaration est un mensonge" – ce genre d'incohérence

utilisé plus tard comme preuve de Gödel et Turing.

(vers 470-410 av. J.-C.) domaine grec

Hippocrate (aucune relation connue avec Hippocrate de Cos,

le célèbre docteur)

a écrit son propreélémentsplus que ça

un siècle avant Euclide. Seuls des fragments survivent, mais ça

Des preuves axiomatiques apparemment similaires à celles d'Euclide

et contient plusieurs des mêmes théorèmes. On dit qu'Hippocrate

ont inventéréduction par absurditéméthode de preuve.

Hippocrate est surtout connu pour son travail sur les trois anciens

dilemmes géométriques: son travail sur le doublage de cubes

(laProblème de Delian) jeter les bases

pour un effort réussi par Archytas et d'autres;

et certains affirment qu'Hippocrate a été le premier à dessiner l'angle général.

Bien sûr, son carré de cercle a finalement échoué, bien que

il a montré des phrases admirables sur "lunes" (en forme de croissant)

fragments de cercle).

Par exemple, la surface de tout triangle rectangle est similaire à celle de tout triangle

la somme des surfaces des deux mottes formées en demi-cercles

est dessinée sur chacun des trois bords du triangle.

Hippocrate a également travaillé avec l'algèbre et l'analyse rudimentaire.(Le doublement des dés et de la section angulaire est souvent appelé

"impossible", mais ils ne sont impossibles que lorsqu'ils sont limités à

boussole effondrée et une règle inhabituelle.

Il existe des solutions ingénieuses disponibles avec d'autres outils.

La construction de l'heptagone habituel est une autre tâche de ce type,

avec des solutions publiées par quatre des hommes de cette liste:

Thabit, Alhazen, Vieta, Conway.)

(vers 420-350 av. J.-C.) domaine grec

Archytas était aussi un homme d'État important

en tant que philosophe. Il a étudié sous le Philolaus de Croton,

était un ami de Platon et guida Eudoxe.

En plus des résultats toujours attribués à lui,

il peut être la source de plusieurs des théorèmes d'Euclide,

et quelques œuvres attribuées à Eudoxe et peut-être à Pythagore.

Récemment, il a été montré pour être magnifiqueProblèmes mécaniques

attribué à (pseudo) Aristote était probablement en fait écrit

par Archytas, faisant de lui l'un des plus grands mathématiciens

de l'antiquité.Archytas a introduit le "mouvement" dans la géométrie, faisant pivoter les courbes

produire des solides.

Si ses écrits avaient survécu, il serait probablement considéré comme un

des géomètres les plus brillants et les plus innovants dans les temps anciens.

Il figure sur la liste des 12 plus grands génies de Cardano.

(Euclide, Aristote, Archimède, Apollonius, Ptolémée,

et le docteur Galen ou Pergame

sont les autres Grecs sur cette liste.)

Le plus célèbre exploit mathématique d'Archita fut

"double cube" (construit un segment de droite plus grand que

une autre racine du cube du facteur de deux).

Bien que d'autres aient résolu le problème

D’autres techniques, la solution d’Archytas pour le doublage de cubes étaient étonnantes car

Cela n’a pas été réalisé dans l’avion, mais a impliqué la traversée

des corps en trois dimensions.

Cette construction (qui a introduitLa courbe d'Archite)

a été appelé "un voyage ces forces de l'imagination spatiale."

Il a inventé le termedes moyens harmoniqueset travaillé avec des moyens géométriques

aussi (prouver que les entiers continus n'ont jamais de moyenne géométrique rationnelle).

Il était un véritable homme-frère: il a élaboré la théorie musicale bien au-delà de Pythagore;

étudié le son, l'optique et la cosmologie;

a inventé la poulie (et un hochet pour occuper les bébés); a écrit à propos de

levier; développé le programme appelé quadrivium;

crédité de trouver la vis;

et doit avoir construit un oiseau en bois à vapeur comme

volé pendant 200 mètres.

Les Archytas sont parfois appelés "le père de la mécanique mathématique".Certains chercheurs pensent que Pythagore et Thalès sont

en partie mythique. Si nous prenons ce point de vue, Archytas (et Hippocrate) sont

devrait être promu dans cette liste.

__
Théétète__ d'Athènes

(417-369 av. J.-C.) Grèce

Théétète est considéré comme le véritable auteur

des livres X et XIII d'Euclideélémentsainsi que certains

travail attribué à Eudoxus. Il était considéré comme l'un des plus brillants

par des mathématiciens grecs, et est le personnage central

dans deux des dialogues de Platon.

C’est Théétète qui a découvert les deux derniers des cinq «solides platoniques»

et a montré qu'il n'y en avait plus.

Il a peut-être été le premier à noter que la racine carrée dequelques-unsentier

Sinon, même un entier doit être irrationnel.

(Le cas √2 est attribué à un élève de Pythagore.)

(408-355 av. J.-C.) domaine grec

Eudoxe a beaucoup voyagé pour son éducation,

malgré ne pas être riche,

étudier les mathématiques avec Archytas à Tarente,

médecine avec Philiston en Sicile,

philosophie avec Platon à Athènes,

continue ses études de mathématiques en Egypte,

visiter la Méditerranée orientale avec ses propres étudiants

et enfin de retour à Cnidus où il s'est établi

en tant qu'astronome, médecin et étiquette.

Ce qui lui est connu est utilisé, à travers les Écritures

euclid et autres, mais il était un

des mathématiciens les plus créatifs de l'ancien monde.Beaucoup de théories

je euclid sélémentsa été prouvé pour la première fois par Eudoxus.

Alors que Pythagore avait été consterné par la découverte de l'irrationalité

Nombres, Eudoxus est connu pour les incorporer dans l’arithmétique.

Il a également développé les premières techniques de l'infinie calorie;

Archimède attribue à Eudoxe la perception d'un principe qui a finalement été appelé

ilsAxiome ou Archimède:

Il évite les paradoxes de Zénon en interdisant les interdictions

infini et sans fin.

Eudoxus fonctionne avec des nombres irrationnels, infinis et des limites

Enfin, les maîtres ont inspiré Dedekind.

Eudoxus a également introduit unL'axiome de la continuité;

il était un pionnier de la géométrie solide;

et il a développé sa propre solution au problème de doublage de cube Delian.

Eudoxe était le premier astronome mathématique majeur;

il développa l'ancienne théorie complexe des orbites planétaires;

et peut-être inventé l'astrolabe.

Il a peut-être inventé le calendrier de 365,25 jours basé sur

ans, bien que ce soit à Jules César de le populariser.

(On dit parfois qu'il le savait

La Terre tourne autour du Soleil, mais cela semble être faux.

C’est au contraire Aristarque de Samos, cité par Archimède, qui

peut être le premier "héliocentriste".)

Menaechmus, un des élèves d’Eudoxe, fut le premier à

Décrivez les parties coniques et utilisez-les pour préparer un disque non platonique.

solution au problème de doublage de cube (et peut-être

problème de cercle-carré aussi).Quatre des découvertes les plus connues d’Eudoxe étaient le volume

d'un cône, extension de l'arithmétique à l'irrationnel, sommation

formule pour les séries géométriques, et

vuecomme limite pour le périmètre polygonal.π

Aucune de ces choses ne semble difficile aujourd'hui, mais cela semble remarquable

qu'ils ont tous d'abord été réalisés par le même homme.

Eudoxus a été cité comme dit

"Je veux doucement brûler à mort comme Phaeton, c'était

le prix pour atteindre le soleil et apprendre sa forme,

sa taille et sa substance. "

Aristote était le scientifique le plus éminent de

vieux monde, et peut-être le philosophe le plus influent

et la logique toujours;

Il est 13ème sur la liste des personnalités les plus influentes de l'histoire de Michael Hart.

Sa science était un programme standard pour près de 2000 ans.

Bien que les sujets physiques n’aient pas pu atteindre

trouver de grands hommes comme Newton et Lavoisier,

Le travail d'Aristote en matière biologique

était super et a servi de paradigme des temps modernes.

Aristote était un guide personnel du jeune Alexandre le Grand.

Le disciple et successeur d'Aristote Théophraste était aussi un

grand chercheur, comme Strato, le successeur de Theophrastus.Bien qu'Aristote soit probablement le plus grand biologiste du monde antique,

Son travail en physique et en mathématiques peut ne pas sembler suffisant

se qualifier pour cette liste.

Mais ses enseignements couvraient un très large éventail et dominaient

le développement de la science ancienne.

Ses écrits sur les définitions, les axiomes et les preuves ont peut-être influencé Euclide;

et il fut l'un des premiers mathématiciens à écrire sur le thème infini.

Ses écrits comprennent des théorèmes géométriques, certains avec des preuves

différent d'Euclid ou absent d'Euclid; un de ces

(vu seulement dans le travail d'Aristote avant Apollonius)

est-ce un cercle est le point si les distances

de deux points donnés sont en relation constante.Aristote est parfois accusé d'avoir tort

les idées freinent le développement de la science.

Mais cette taxe est injuste; Aristote lui-même a souligné l'importance

d'observation et d'expérimentation, et être prêt à rejeter

vieilles hypothèses et en préparer de nouvelles.

Et bien que les théories géométriques d’Aristote, bien acceptées, soient unanimes

N'était-ce pas son propre travail, son statut de plus influent

logique et philosophe à travers l'histoire

fait de lui un candidat fort pour la liste.

(environ 322-275 av. J.-C.) Grèce / Égypte

Euclide d'Alexandrie (à ne pas confondre avec l'étudiant de Socrate,

Euclide de Mégare, qui vivait un siècle plus tôt),

dirigé l'école de mathématiques dans son ensemble

Université d'Alexandrie. Quelque chose d'autre est probablement connu pour sa vie,

Mais plusieurs réalisations mathématiques très importantes lui sont imputées.

Il fut le premier à prouver que c'était sans fin

beaucoup de nombres premiers; il a produit des preuves incomplètes

Le théorème unique de factorisation (théorème fondamental de l'arithmétique);

et il a évoluéAlgorithme d'Euclidecalculer gcd.

Il a présenté les nombres premiers de Mersenne

et observé(M^{2}+ M) / 2

est toujours parfait (sous la forme de Pythagore) siMc'est Mersenne.

(L'inverse est qu'un nombre parfait a un nombre similaire

Mersenne prime, a été géré par Alhazen et prouvé par Euler.)

Ses livres contiennent de nombreux théorèmes bien connus, mais une grande partieéléments

était dû à des prédécesseurs comme Pythagore (la plupart des livres I et II),

Hippocrate (livre III), Theodorus, Eudoxus (livre V),

Archytas (peut-être livre VIII) et Theaetet.

Livre je commence avec une preuve élégante de la construction de compas rigide

peut être mis en œuvre avec une boussole simultanée.

(Étant donné A, B, C, trouvez CF = AB au début

construire un triangle équilatéral ACD;

utilisez la boussole pour trouver E sur AD avec AE = AB;

et enfin trouver F sur DC avec DF = DE.)

Bien que les notions de trigonométrie soient utilisées, les théorèmes d’Euclide étaient

inclure certains étroitement liés aux lois des sinus et des cosinus.

Parmi plusieurs livres attribués à Euclid se trouve

La division de(une discussion mathématique de la musique),

échelletélescopes,

La cartoptrique(une thèse sur la théorie des miroirs),

un livre sur la géométrie sphérique, un livre sur les erreurs logiques,

et son vaste manuel de mathématiquesles éléments.

Plusieurs de ses chefs-d'œuvre ont été perdus, y compris

Fonctionne sur les sections coniques et autres sujets géométriques avancés.

Apparemment Desargues et le théorème homosexuel (un couple de triangles

est coaxial si et seulement si il est prouvé de manière copolaire dans un

de ces œuvres perdues; C'est la méthode théorique de base

qui a commencé l'étude de la géométrie projective.

Euclid se classe 14ème sur la célèbre liste de Michael Hart

Les personnes les plus influentes de l'histoire.

les élémentsintroduit le concept d'axiome et de théorème;

a été utilisé comme manuel pendant 2000 ans; et est en fait toujours la base

pour la géométrie du lycée, faites

Euclid le principal professeur de mathématiques tout le temps.

Certains croient que sa meilleure inspiration était de reconnaître que

Parallèlement à Postulat doit être un axiome plutôt qu'une phrase.Il y a beaucoup de citations célèbres sur Euclid et ses livres.

Abraham Lincoln a quitté son étude de droit quand il n'a pas

sais ce que "démontrer" signifiait et "rentrait chez moi chez mon père

(à lire Euclid), et y resta jusqu'à ce que je puisse donner une suggestion

dans les six livres d'euclide en vue.

Ensuite, j'ai découvert ce qui démontre des moyens et je suis retourné à mes études de droit. "

(287-212 av. J.-C.) domaine grec

Archimède est universellement reconnu pour être

le plus grand des vieux mathématiciens.

Il a étudié à l’école d’Euclide (probablement après

La mort d'Euclide, mais son travail a dépassé de loin, et a même dépassé,

Le travail d'Euclide.

(Par exemple, certaines des phrases les plus difficiles d'Euclide sont simples

Conséquences analytiques du lemma d’Archimède des centroïdes.

Ses réalisations sont particulièrement impressionnantes compte tenu de

manque de bonne notation mathématique à son époque.

Son témoignage est répertorié non seulement pour son éclat, mais aussi pour

clarté incomparable, avec un cinéma moderne (Heath) décrivant

Les thèses d'Archimède qui "sans exception des monuments de mathématiques

exposition … si impressionnant dans leur perfection que créer un

se sent accompagné par la crainte dans l'esprit du lecteur. "

Archimède a fait des progrès dans la théorie des nombres, l'algèbre et l'analyse,

mais est surtout connu pour ses nombreuses théories

des aéronefs et de la géométrie solide.

Il aurait pu être le premier à prouver la formule de Heron

zone d'un triangle.

Son excellente approche de√3l'indique

il s'était en partie attendu à la méthode des factions continues.

Il a développé une méthode récursive pour représenter les grands

entiers, et a été le premier à remarquer la loi des exposants,

10.^{un}· 10^{b}= 10^{a + b}

Il a travaillé avec des exposants et a développé des notes et des noms simples.

pour les nombres supérieurs à 10 ^ (10 ^ 16); Cela semblera plus sensationnel

Quand vous vous en souvenez, c'était encore 18 siècles avant les Européens

serait inventer le mot "millions".Archimède a trouvé une méthode pour faire pivoter un angle arbitraire (en utilisant

unmarquablesrègle – la construction est impossible

utilise des règles strictement platoniques).

Un de ses résultats géométriques les plus notables et célèbres

était de déterminer la surface d'une section parabolique, pour laquelle

il a offert deux preuves indépendantes, une

utiliser sonLe principe de l'événement,

l'autre utilise une série géométrique.

Une partie du travail d'Archimède survit simplement parce que Thabit ibn Qurra

traduit autrement perduLivre de lemmes; Il contient

méthode de trisection angulaire et plusieurs théorèmes ingénieux

sur les cercles inscrits.

(Thabit montre comment construire un heptagone régulier, ça ne peut pas être

préciser si cela venait d'Archimède ou était caractérisé par Thabit

en étudiant la méthode angle-trisection d'Archimède.)

Parmi les autres résultats connus uniquement comme utilisés

ilsSolides semi-régulaires archimédiensrapporté par Pappus,

et Théorème des accords brisés rapporté par Alberuni.Archimède et Newton peuvent être les deux meilleures géométries de tous les temps, bien que

Bien que chacun produise de brillantes preuves géométriques, ils l'utilisent souvent

calcul non strict àdécouvrirrésultats, puis préparé

preuve géométrique stricte de publication.

Il a utilisé la calculatrice intégrée

pour déterminer les points médians de l'hémisphère et cylindrique

coin et le volume de deux croix de cylindre.

Il a également travaillé avec diverses spirales, paraboloïdes révolutionnaires, etc.

Bien que Archimède n'ait pas développé de différenciation (intégration inverse)

Michel Chasles le reconnaît (avec Kepler, Cavalieri et Fermat,

qui vécurent tous plus de 18 siècles plus tard)

comme l'un des quatre qui ont développé la calculatrice avant Newton et Leibniz.

(Bien qu'il connaisse l'aide de l'infini, il est

accepté "Théorème d'Eudoxe" qui leur interdit d'éviter

Les paradoxes de Zeno. Les mathématiciens modernes se réfèrent à l'objectif théorique

Axiome d’Archimède.)Archimède était un astronome (détails de ses découvertes perdues,

mais il est probable qu'il savait que la terre tournait autour du soleil.

Il fut l'un des plus grands mécaniciens jamais découvert

Archimedes "Principe hydrostatistique (un corps partiellement ou totalement

immergé dans un liquide perd effectivement un poids égal

le poids du liquide se déplace).

Il a développé les fondements mathématiques qui le sous-tendent

Avantage des machines de base: poulie à levier, à vis et composite.

Bien que la vis puisse avoir été inventée par Archytas,

et l'homme de l'âge de pierre (et même d'autres animaux) a utilisé le levier,

On dit que la connexion

La tranche a été inventée par Archimède lui-même.

Pour ces réalisations, il est souvent classé devant Maxwell

Être appelé l'un des trois plus grands physiciens de tous les temps.

Archimède était un inventeur productif:

en plus de percevoir la poulie composite, il

a inventé la pompe à vis hydraulique (appelée vis d'Archimède);

un planétarium miniature; et plus

krigsmaskiner – katapult, parabolske speil for å brenne fiendtlige skip,

en dampkanon og 'The Arch of Archimedes'.

Noen lærde tilskriverAntikythera mekanisme

til Archimedes. (Is it the Archimedean planetarium mentioned by Cicero?)His books include

Floating Bodies,

Spirals,

The Sand Reckoner,

Measurement of the Circle,

Sphere and Cylinder,

Plane Equilibriums,

Conoids and Spheroids,

Quadrature of Parabola,

The Book of Lemmas(translated and attributed by Thabit ibn Qurra),

various now-lost works (on Mirrors, Balances and Levers, Semi-regular Polyhedra,

etc.) cited by Pappus or others,

and (discovered only recently,

and often called his most important work)The Method.

He developed theStomachionpuzzle (and solved a difficult

enumeration problem involving it); other famous gems

inkludereThe Cattle-Problem.

The Book of Lemmascontains various geometric gems

("the Salinon," "the Shoemaker's Knife", etc.) and is credited

to Archimedes by Thabit ibn Qurra but the attribution is disputed.Archimedes discovered formulae for the volume and surface area

of a sphere, and may even have been first to notice and prove the

simple relationship between a circle's circumference and area.

For these reasons,

πis often calledArchimedes' constant.

His approximation223/71

was the best of his day.

(Apollonius soon surpassed it, but by using Archimedes' method.)

Archimedes' Equiarea Map Theorem asserts that a sphere and its enclosing

cylinder have equal surface area (as do the figures' truncations).

Archimedes also proved that the volume of that sphere is two-thirds the volume

of the cylinder.

He requested that a representation of such a sphere and

cylinder be inscribed on his tomb.

That Archimedes shared the attitude of later mathematicians like

Hardy and Brouwer is suggested by Plutarch's comment that Archimedes regarded

applied mathematics "as ignoble and sordid … and did not deign

to (write about his mechanical inventions; instead)

he placed his whole ambition in those speculations the beauty and subtlety of

which are untainted by any admixture of the common needs of life."Some of Archimedes' greatest writings (including

The MethodetFloating Bodies) are preserved only on a

palimpsest rediscovered in 1906 and mostly deciphered only after 1998.

Ideas unique to that work are an anticipation of Riemann

integration, calculating

the volume of a cylindrical wedge (previously first attributed to Kepler);

along with Oresme and Galileo he was among the few to comment

on the "equinumerosity paradox" (the fact that are as many perfect

squares as integers).

Although Euler and Newton

may have been the most important mathematicians,

and Gauss, Weierstrass and Riemann the greatest theorem provers,

it is widely accepted that

Archimedes was the greatest genius who ever lived.

Yet, Hart omits him altogether from his list of Most Influential Persons:

Archimedes was simplyaussifar ahead of his time to have great historical

significance.

(Some think the Scientific Revolution would have begun sooner

avaitThe Methodbeen discovered four or five centuries earlier.

vous kan

read a 1912 translation of parts ofThe Methodon-line.)

Eratosthenes was one of the greatest polymaths; he is called

the Father of Geography, was Chief Librarian at Alexandria, was a

poet, music theorist,

mechanical engineer (anticipating laws of elasticity, etc.),

astronomer (he is credited as first to measure the circumference

of the Earth), and an outstanding mathematician.

He is famous for his prime number Sieve, but more impressive was his

work on the cube-doubling problem which he related to the design

of siege weapons (catapults) where a cube-root calculation is needed.Eratosthenes had the nickname

Beta; he was a master of several

fields, but was only second-best of his time.

His better was also his good friend:

Archimedes of Syracuse dedicatedThe Methodto Eratosthenes.

Apollonius Pergaeus, called "The Great Geometer,"

is sometimes considered the second greatest

of ancient Greek mathematicians. (Euclid, Eudoxus and Archytas

are other candidates for this honor.)

His writings on conic sections have been studied until

modern times;

he developed methods for normals and curvature.

(He is often credited with inventing the names for parabola,

hyperbola and ellipse; but these shapes were previously described

by Menaechmus, and their names may also predate Apollonius.)

Although astronomers eventually concluded it was not physically correct,

Apollonius developed the "epicycle and deferent" model of planetary orbits,

and proved important theorems in this area.

He deliberately emphasized the beauty of pure, rather than

applied, mathematics, saying his theorems were

"worthy of acceptance for the sake of the demonstrations themselves."

The following

generalization of the Pythagorean Theorem, where M is the midpoint of BC,

is called Apollonius' Theorem:

AB.^{ 2}+ AC^{ 2}=

2(AM^{ 2}+ BM^{ 2})Many of his works have survived only in a fragmentary form,

and the proofs were completely lost.

Most famous was theProblem of Apollonius,

which is to find a circle tangent to three objects, with the

objects being points, lines, or circles, in any combination.

Constructing the eight circles each tangent to three other circles

is especially challenging, but just finding the two circles

containing two given points and tangent to a given line is

a serious challenge.

Vieta was renowned for discovering methods for all ten

cases of this Problem.

Other great mathematicians who have enjoyed reconstructing

Apollonius' lost theorems

include Fermat, Pascal, Newton, Euler, Poncelet and Gauss.In evaluating the genius of the ancient Greeks,

it is well to remember that their achievements were made

without the convenience of modern notation.

It is clear from his writing that Apollonius almost developed

the analytic geometry of Descartes, but

failed due to the lack of such elementary concepts as negative numbers.

Leibniz wrote "He who understands Archimedes and Apollonius will admire less

the achievements of the foremost men of later times."

Chinese mathematicians excelled for thousands of years,

and were first to discover various algebraic and geometric principles.

There is some evidence that Chinese writings influenced India and

the Islamic Empire, and thus, indirectly, Europe.

Although there were great Chinese mathematicians a thousand years before

the Han Dynasty (as evidenced by the ancientZhoubi Suanjing)

and innovations continued for centuries after Han,

the textbookNine Chapters on the Mathematical Art

has special importance.

Nine Chapters(known in Chinese asJiu Zhang Suan Shu

ouChiu Chang Suan Shu)

was apparently written during the early Han Dynasty (about 165 BC)

by Chang Tshang (also spelled Zhang Cang).Many of the mathematical concepts of the early Greeks were

discovered independently in early China.

Chang's book gives methods of arithmetic (including cube roots)

and algebra,

uses the decimal system (though zero was represented as just a space,

rather than a discrete symbol),

proves the Pythagorean Theorem,

and includes a clever geometric proof that the perimeter of

a right triangle times the radius of its inscribing

circle equals the area of its circumscribing rectangle.

(Some of this may have been added after the time of Chang;

some additions attributed to Liu Hui are mentioned in his mini-bio;

other famous contributors are Jing Fang and Zhang Heng.)

Nine Chapterswas probably based on earlier books,

lost during the great book burning of 212 BC, and

Chang himself may have been a lord who commissioned others to

prepare the book.

Moreover, important revisions and commentaries were added

after Chang, notably by Liu Hui (ca 220-280).

Although Liu Hui mentions Chang's skill, it isn't clear

Chang had the mathematical genius to qualify for this list,

but he would still be a strong candidate due to his book's

immense historical importance:

It was the dominant Chinese mathematical text for centuries,

and had great influence throughout the Far East.

After Chang,

Chinese mathematics continued to flourish, discovering

trigonometry, matrix methods, the Binomial Theorem, etc.

Some of the teachings made their way to India,

and from there to the Islamic world and Europe.

There is some evidence that the Hindus borrowed the

decimal system itself from books likeNine Chapters.No one person can be credited with the invention of

the decimal system, but key roles were played by early Chinese

(Chang Tshang and Liu Hui),

Brahmagupta (and earlier Hindus including Aryabhata),

and Leonardo Fibonacci.

(After Fibonacci, Europe still did not embrace the decimal system

until the works of Vieta, Stevin, and Napier.)

**Hipparque** of Nicaea and Rhodes

Ptolemy may be the most famous astronomer

before Copernicus, but he borrowed heavily from Hipparchus,

who should thus be considered (along with Galileo and Edwin Hubble)

to be one of the three greatest astronomers ever.

Careful study of thefeilin the catalogs

of Ptolemy and Hipparchus reveal both that Ptolemy

borrowed his data from Hipparchus, and that Hipparchus

used principles of spherical trig to simplify his work.

Classical Hindu astronomers, including the 6th-century genius

Aryabhata, borrow much from Ptolemy and Hipparchus.Hipparchus is called the "Father of Trigonometry"; han

developed spherical trigonometry,

produced trig tables, and more.

He produced at least fourteen texts of physics and mathematics

nearly all of which have been lost, but which seem to

have had great teachings, including

much of Newton's Laws of Motion.

In one obscure surviving work he demonstrates familiarity

with the combinatorial enumeration method now called Schröder's Numbers.

He invented the circle-conformal stereographic

and orthographic map projections which carry his name.

As an astronomer, Hipparchus is credited with the discovery

of equinox precession, length of the year, thorough

star catalogs, and invention

of the armillary sphere and perhaps the astrolabe.

He had great historical influence in Europe, India and

Persia, at least if credited also with Ptolemy's influence.

(Hipparchus himself was influenced by Babylonian astronomers.)

Hipparchus' work implies a better approximation toπ

than that of Apollonius, perhaps it wasπ ≈ 377/120qui

Ptolemy used.ils

Antikythera mechanismis an astronomical clock

considered amazing for its time.

It may have been built about the time of Hipparchus'

death, but lost after a few decades

(remaining at the bottom of the sea for 2000 years).

The mechanism implemented the complex orbits which Hipparchus had developed

to explain irregular planetary motions;

it's not unlikely the great genius helped design this intricate

analog computer, which may have been built in Rhodes

where Hipparchus spent his final decades.

(Recent studies suggest that the mechanism was designed

in Archimedes' time, and that therefore that genius

might have been the designer.)(Let's mention another Greek astronomer contemporaneous to Hipparchus,

Seleucus of Seleucia (ca 190-145 BC), who is noted for supporting

heliocentrism. explaining tides, and proposing that the universe is infinite.)

__of Alexandria__

Heron (or Hero) was apparently a teacher at the great university

of Alexandria, but there is much uncertainty about his life and work.

He wrote on mechanics (analysing the five basic machines of

mechanical advantage), astronomy (determining longitudes),

hydrostatics, architecture, surveying, optics (he introduced the

'shortest-distance' explanation for mirror reflection),

arithmetic (finding square roots and cube roots),

and geometry (finding the areas and volumes of various shapes).

He was an inventor; he was first to describe a syringe, a windmill,

a pump for extinguishing fires, and some very primitive counters and computers.

He is especially famous for his invention of the aeolipile which

rotated using

steam from an attached cauldron, and is considered the first steam engine.

(Vitruvius may have described such a machine before Heron.)

He is noted for designing various toys (probably developed as teaching

aids for his lectures); these included a puppet theater driven by strings

and weights, a robot trumpet, trick wine glasses, a windmill-driven organ,

and coin-operated vending machine.

His most famous discovery in mathematics was

Heron's Formula for the area A of a triangle with sides a,b,c:

FR^{2}= s(s-a)(s-b)(s-c) where s = (a+b+c)/2But there is some controversy about the actual authorship of Heron's books;

and much of Heron's best physics and mathematics (possibly

including Heron's Formula) appear to repeat discoveries by Archimedes.

Thus, despite his fame, we do not include Heron on our List.

__of Alexandria__

Menelaus wrote several books on geometry and

trigonometry, mostly lost except for his works on solid geometry.

His work was cited by Ptolemy, Pappus, and Thabit;

especially the Theorem of Menelaus itself which is a

fundamental and difficult theorem very useful in projective geometry.

He also contributed much to spherical trigonometry.

Disdaining indirect proofs (anticipating later-day constructivists)

Menelaus found new, more fruitful proofs for several of Euclid's results.

Tiberius(?) Claudius **Ptolemaeus**

__of Alexandria__

(ca 90-168) Egypt (in Greco-Roman domain)

Ptolemy, the Librarian of Alexandria,

was one of the most famous of ancient Greek scientists.

His textbooks were among the most important of the ancient world,

perhaps because they supplanted most that had come before.

He provided new insights into optics and was the best geographer of

his day.

Among his mathematical results, most famous may be Ptolemy's Theorem

(AC·BD=AB·CD+BC·AD

if and only ifABCDis a cyclic quadrilateral).

This theorem has many useful corollaries; it was frequently applied

in Copernicus' work.

Ptolemy wrote on trigonometry, optics, geography, and map projections;

but is most famous for his astronomy,

where he perfected the geocentric model of planetary motions.

For this work, Cardano included Ptolemy on his List of 12 Greatest Geniuses,

but removed him from the list after learning of Copernicus' discovery.

Interestingly, Ptolemy wrote that the fixed point in a model of planetary

motion was arbitrary, but rejected the Earth spinning on its axis

since he thought this would lead to powerful winds.

Ptolemy discussed and tabulated the 'equation of time,' documenting the

irregular apparent motion of the Sun. (It took fifteen centuries

before this irregularity was correctly attributed to Earth's elliptical orbit.)

The mystery of celestial motions directed scientific inquiry

for thousands of years.

Except for some Pythagoreans like Philolaus of Croton,

thinkers generally assumed that the Earth was the center of the universe,

but this made it very difficult to explain the orbits of the other planets.

This problem had been considered by Eudoxus, Apollonius, and Hipparchus,

who developed a very complicated geocentric model involving

concentric spheres and epicycles.

Ptolemy perfected (or, rather, complicated) this model even further,

introducing 'equants' to further fine-tune the orbital speeds;

this model was the standard for 14 centuries.

While some Greeks, notably Aristarchus and Seleucus,

proposed heliocentric models,

these were rejected because there was no parallax among stars.

(Aristarchus guessed that the stars were at an almost unimaginable

distance, explaining the lack of parallax.

Aristarchus would be almost unknown except that

Archimedes mentions, and assumes, Aristarchus' heliocentrism in

The Sand Reckoner.

I suspect that Archimedes accepted heliocentrism, but thought saying

so openly would distract from his work.

Several thinkers proposed a hybrid system with Mercury and Venus rotating

the Sun but the outer planets and the Sun itself rotating Earth; disse

thinkers may have included ancient Egyptians, the Greek Heraclides of Pontus,

some of the Islamic scientists,

a member of Madhava's Kerala school, and Tycho Brahe — the astronomer

who linked Copernicus to Kepler.A related question is: Does the Earth spin daily on its axis? tous

heliocentrists, beginning with Heraclides of Pontus, seem to have accepted

that, as well as some who were more doubtful

about the Earth's annual orbit.

And another related question is: Is the universe finite, or is it infinite?

Democritus, Seleucus, Nicholas of Cusa and Giordano Bruno

were three who proposed an infinite universe prior to Galileo.Hipparchus was another ancient Greek who considered heliocentrism but,

because he never guessed

that orbits were ellipses rather than cascaded circles, was unable to come

up with a heliocentric model that fit his data.)

Aryabhata, Alhazen, Alberuni, Omar Khayyám, (perhaps some other

Islamic mathematicians like al-Tusi), Regiomontanus, and Leonardo

da Vinci are

other great pre-Copernican mathematicians who may have accepted

the possibility of heliocentrism.

Another reason to doubt that the Earth moves, is that we don't

feel that motion, or see its effect on falling bodies. cette

difficulty, which almost disappears once Newton's First Law of Motion

is accepted, was addressed before Newton by Jean Buridan,

Nicole Oresme, Giordano Bruno,

Pierre Gassendi (1592-1655) and, of course, Galileo Galilei.The great skill demonstrated by Ptolemy and his predecessors in

developing their complex geocentric cosmology

may haveset backscience since in fact

the Earth rotates around the Sun.

The geocentric models couldn't

explain the observed changes in the brightness of Mars or Venus,

but it was the phases of Venus, discovered by Galileo after the invention

of the telescope, that finally led to general acceptance of heliocentrism.

(Ptolemy's model predicted phases, but timed quite differently from

Galileo's observations.)Since the planets move without friction, their motions offer a pure

view of the Laws of Motion; this is one reason that the heliocentric

breakthroughs of Copernicus, Kepler and Newton triggered the advances

in mathematical physics which led to the Scientific Revolution.

Heliocentrism offered an even more key understandingthat lead to

massive change in scientific thought.

For Ptolemy and other geocentrists, the "fixed" stars

were just lights on a sphererotating around the earth, but after

the Copernican Revolution the fixed stars were understood to be

immensely far away; this made it possible to imagine that they were

themselves suns, perhaps with planets of their own. (Nicole Oresme

and Nicholas of Cusa were pre-Copernican thinkers who wrote on both

the geocentric question and the possibility of other worlds.)

The Copernican perspective led

Giordano Bruno and Galileo to posit a single common set of

physical laws which ruled both on Earth and in the Heavens.

(It was this, rather than just the happenstance of planetary orbits,

that eventually most outraged the Roman Church….

And we're getting ahead of our story:

Copernicus, Bruno, Galileo and Kepler lived 14 centuries after Ptolemy.)

Liu Hui made major improvements

to Chang's influential textbookNine Chapters,

making him among the most important of Chinese mathematicians ever.

(He seems to have been a much better mathematician

than Chang, but just as Newton might have gotten nowhere without Kepler,

Vieta, Huygens, Fermat, Wallis, Cavalieri, etc., so Liu Hui might

have achieved little had Chang not preserved the ancient

Chinese learnings.)

Among Liu's achievements

are an emphasis on generalizations and proofs,

incorporation of negative numbers into arithmetic,

an early recognition of the notions of infinitesimals and limits,

the Gaussian elimination method of solving

simultaneous linear equations,

calculations of solid volumes (including the use of Cavalieri's Principle),

anticipation of Horner's Method,

and a new method to calculate square roots.

Like Archimedes, Liu discovered the formula for a circle's area;

however he failed to calculate a sphere's volume, writing

"Let us leave this problem to whoever can tell the truth."Although it was almost child's-play for any of them,

Archimedes, Apollonius, and Hipparchus had all improved

precision ofπ's estimate.

It seems fitting that Liu Hui did join that select company of

record setters: He developed a recurrence formula for

regular polygons allowing arbitrarily-close approximations

àπ.

He also devised an interpolation formula to simplify

that calculation; this yielded the "good-enough" value 3.1416,

which is still taught today in primary schools.

(Liu's successors in China included Zu Chongzhi, whoa fait

determine sphere's volume, and whose approximation forπ

held the accuracy record for nine centuries.)

Diophantus was one of the most influential

mathematicians of antiquity; he wrote several books on

arithmetic and algebra,

and explored number theory further than anyone earlier.

He advanced a rudimentary arithmetic and algebraic notation, allowed

rational-number solutions to his problems rather than just integers,

and was aware of results like the Brahmagupta-Fibonacci Identity;

for these reasons he is often called the "Father of Algebra."

His work, however, may seem quite limited to a modern eye:

his methods were not generalized, he knew nothing

of negative numbers, and, though he often dealt with quadratic

equations, never seems to have commented on their second solution.

His notation, clumsy as it was, was used for many centuries.

(The shorthandx^{3}for "x cubed" was not invented until Descartes.)

Very little is known about Diophantus (he might even have come from

Babylonia, whose algebraic ideas he borrowed).

Many of his works have been lost, including proofs for lemmas

cited in the surviving work, some of which are so

difficult it would almost stagger the imagination to

believe Diophantus really had proofs.

Among these are Fermat's conjecture (Lagrange's theorem)

that every integer is the sum of four squares, and the

following:

"Given any positive rationalsun,baveca>b,

there exist positive rationalsc,résuch that

un".^{3}-b^{3}= c^{3}+d^{3}

(This latter "lemma" was investigated by Vieta and Fermat and

finally solved, with some difficulty, in the 19th century.

It seems unlikely that Diophantus actually had proofs for such "lemmas.")

Pappus, along with Diophantus, may have been

one of the two greatest Western mathematicians

during the 13 centuries that separated Hipparchus and Fibonacci.

He wrote about arithmetic methods, plane and solid geometry,

the axiomatic method, celestial motions and mechanics.

In addition to his own original research, his texts are

noteworthy for preserving works of earlier mathematicians

that would otherwise have been lost.Pappus' best and most original result, and the one which gave

him most pride, may be the Pappus Centroid theorems

(fundamental, difficult and powerful theorems of solid geometry

later rediscovered by Paul Guldin).

His other ingenious geometric theorems

include Desargues' Homology Theorem (which Pappus attributes

to Euclid), an early form of Pascal's Hexagram Theorem,

called Pappus' Hexagon Theorem and related to a fundamental

theorem: Two projective pencils

can always be brought into a perspective position.

For these theorems, Pappus is sometimes called

the "Father of Projective Geometry."

Pappus also demonstrated how to perform angle trisection and

cube doubling if one can use mechanical curves like

a conchoid or hyperbola.

He stated (but didn't prove) theIsoperimetric Theorem, also

writing "Bees know this fact which is useful to them,

that the hexagon … will hold more honey

for the same material than (a square or triangle)."

(That a honeycomb partition minimizes material for an equal-area

partitioning was finally proved in 1999 by Thomas Hales, who also

proved the related Kepler Conjecture.)

Pappus stated, but did not fully solve, theProblem of Pappus

which, given an arbitrary collection of lines in the plane, asks for

the locus of points whose distances to the lines have

a certain relationship.

This problem was a major inspiration for Descartes and was finally

fully solved by Newton.For preserving the teachings of Euclid and Apollonius,

as well as his own theorems of geometry, Pappus certainly

belongs on a list of great ancient mathematicians.

But these teachings lay dormant during Europe's Dark Ages, diminishing

Pappus' historical significance.

**Mathematicians after Classical Greece**

Alexander the Great spread Greek culture to Egypt and much of the Orient;

thus even Hindu mathematics may owe something to the Greeks.

Greece was eventually absorbed into the Roman Empire

(with Archimedes himself famously killed by a Roman soldier).

Rome did not pursue pure science as Greece had (as we've seen,

the important mathematicians of the Roman era

were based in the Hellenic East)

and eventually Europe fell into a Dark Age.

The Greek emphasis on pure mathematics and proofs was

key to the future of mathematics, but they were missing

an even more important catalyst: a decimal place-value system

based on zero and nine other symbols.

**Decimal system — from India? China?? Persia???**

Laplace called the decimal system "a profound and important idea

(given by India) which

appears so simple to us now that we ignore its true merit … in

the first rank of useful inventions (but) it escaped the

genius of Archimedes and Apollonius."

But even after Fibonacci introduced the system to Europe, it was another

400 years before it came into common use.Ancient Greeks, by the way, did not use the

unwieldy Roman numerals, but rather used 27 symbols, denoting

1 to 9, 10 to 90, and 100 to 900.

Unlike our system, with ten digits separate from the alphabet,

the 27 Greek number symbols were themêmeas their

alphabet's letters; this might have hindered

the development of "syncopated" notation.

The most ancient Hindu records did not use the

ten digits of Aryabhata, but rather a system

similar to that of the ancient Greeks, suggesting that

China, and not India, may indeed be the "ultimate" source of the modern

decimal system.The Chinese used a form of decimal abacus as early as 3000 BC;

if it doesn't qualify, by itself, as a "decimal system" then

pictorial depictions of its numbers would.

Yet for thousands of years after its abacus, China had no zero symbol

other than plain space; and apparently didn't have one until after

the Hindus. Ancient Persians and Mayansa faithave place-value

notationaveczero symbols, but neither qualify as inventing

a base-10 decimal system: Persia used the base-60 Babylonian system;

Mayans used base-20.

(Another difference is that the Hindus had nine distinct digit symbols

to go with their zero, while earlier place-value systems

built up from just two symbols: 1 and either 5 or 10.)

The Old Kingdom Egyptians did use a base-ten system, but it was

similar to that of Greece and Vedic India:

1, 10, 100 were depicted as separate symbols.Conclusion: The decimal place-value system with zero symbol seems

to be an obvious invention that in fact was very hard to invent.

If you insist on a single winner then India might be it.

But China, Babylonia, Persia and even the Mayans deserve Honorable Mention!

(476-550) Ashmaka & Kusumapura (India)

Indian mathematicians excelled for thousands of years,

and eventually even developed advanced techniques like Taylor series

before Europeans did, but they are

denied credit because of Western ascendancy.

Among the Hindu mathematicians, Aryabhata (called Arjehir by Arabs)

may be most famous.While Europe was in its early "Dark Age,"

Aryabhata advanced arithmetic, algebra,

elementary analysis, and especially

plane and spherical trigonometry, using the decimal system.

Aryabhata is sometimes called the "Father of Algebra"

instead of al-Khowârizmi (who himself cites the work of Aryabhata).

His most famous accomplishment in mathematics

was theAryabhata Algorithm(connected

to continued fractions) for solving Diophantine equations.

Aryabhata made several important discoveries in astronomy,

e.g. the nature of moonlight, and concept of sidereal year;

his estimate of the Earth's circumference was more accurate than

any achieved in ancient Greece.

He was among the very few ancient scholars who realized the Earth

rotated daily on an axis; claims that he also espoused heliocentric

orbits are controversial, but may be confirmed by the writings of al-Biruni.

Aryabhata is said to have introduced the constante.

He usedπ ≈ 3.1416;

it is unclear whether he discovered this independently

or borrowed it from Liu Hui of China.

Although it was first discovered by Nicomachus three centuries earlier,

Aryabhata is famous for the identity

Σ (k^{3}) = (Σ k)^{2}Some of Aryabhata's achievements, e.g. an excellent approximation

to the sine function, are known only from the writings of

Bhaskara I, who wrote: "Aryabhata is

the master who, after reaching the furthest shores and plumbing the inmost

depths of the sea of ultimate knowledge of mathematics, kinematics

and spherics, handed over the three sciences to the learned world."

**Brahmagupta** `Bhillamalacarya'

No one person gets unique credit for the invention of the

decimal system but Brahmagupta's textbookBrahmasphutasiddhanta

was very influential, and is sometimes considered

the first textbook "to treat zero as a number in its own right."

It also treated negative numbers.

(Others claim these were first seen 800 years earlier in

Chang Tshang's Chinese text and were implicit in

what survives of earlier Hindu works, but Brahmagupta's

text discussed them lucidly.)

Along with Diophantus, Brahmagupta was also among the first to express

equations with symbols rather than words.Brahmagupta Bhillamalacarya (`The Teacher from Bhillamala')

made great advances in arithmetic,

algebra, numeric analysis, and geometry.

Several theorems bear his name, including

the formula for the area of a cyclic quadrilateral:

16 A^{2}= (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)

Another famous Brahmagupta theorem dealing with

such quadrilaterals can be phrased

"In a circle, if the chords AB and CD are perpendicular

and intersect at E, then the line from E which bisects AC

will be perpendicular to BD."

He also began the study of rational quadrilaterals which Kummer would

eventually complete.

Proving Brahmagupta's theorems are good challenges even today.In addition to his famous writings on practical mathematics

and his ingenious theorems of geometry,

Brahmagupta solved the general quadratic equation,

and worked on number theory problems.

He was first to find a general solution to the simplest Diophantine

form.

His work on Pell's equations has been called "brilliant"

and "marvelous."

He proved the Brahmagupta-Fibonacci Identity

(the set of sums of two squares is closed under multiplication).

He applied mathematics to astronomy, predicting eclipses, etc.

The astronomer Bháskara I, who takes the suffix "I"

to distinguish him from

the more famous Bháskara who lived five centuries later,

made key advances to the positional decimal number notation,

and was the first known to use the zero symbol.

He preserved some of the teachings of Aryabhata which would

otherwise have been lost; these include a famous

formula giving an excellent approximation to the

sinfunction, as well as, probably, the zero symbol itself.Among other original contributions to mathematics,

Bháskara I was first to state Wilson's Theorem (which should

perhaps be called Bháskara's Conjecture):

(n-1)! ≡ -1 (mod n)if and only if n is prime

Bháskara's Conjecture was rediscovered by Alhazen, Fibonacci,

Leibniz and Wilson. The "only if" is easy but the difficult "if"

part was finally proved by Lagrange in 1771.

Since Lagrange has so many other Theorems named

after him, Bháskara's Conjecture is always called "Wilson's Theorem."

__Muhammed `Abu Jafar' ibn Musâ al-Khowârizmi__

(ca 780-850) Khorasan (Uzbekistan), Iraq

Al-Khowârizmi (aka Mahomet ibn Moses) was a Persian

who worked as a mathematician, astronomer and geographer

early in the Golden Age of Islamic science.

He introduced the Hindu decimal system to the Islamic world and Europe;

invented the horary quadrant; improved the sundial;

developed trigonometry tables; and improved on Ptolemy's

astronomy and geography.

He wrote the book

Al-Jabr, which demonstrated simple algebra and geometry,

and several other influential books.

Unlike Diophantus' work, which dealt in specific examples,

Al-Khowârizmi was the first algebra text to present general methods;

he is often called the "Father of Algebra."

(Diophantus did, however, use superior "syncopated" notation.)

The wordalgorithmis borrowed from Al-Khowârizmi's name,

etalgebrais taken from the name of his book.

He also coined the wordcipher, which became Englishnull

(although this was just a translation from the Sanskrit word

for zero introduced by Aryabhata).

He was an essential pioneer for Islamic science,

and for the many Arab and Persian mathematicians who followed;

and hence also for Europe's eventual Renaissance which was heavily dependent

on Islamic teachings.

Al-Khowârizmi's texts on algebra and decimal arithmetic are

considered to be among the most influential writings ever.

__Ya'qub `Abu Yusuf' ibn Ishaq al-Kindi__

Al-Kindi (called Alkindus or Achindus in the West)

wrote on diverse philosophical subjects, physics, optics,

astronomy, music, psychology, medicine, chemistry, and more.

He invented pharmaceutical methods, perfumes, and distilling of alcohol.

In mathematics, he popularized the use of the decimal system,

developed spherical geometry, wrote on many other topics and was

a pioneer of cryptography (code-breaking).

(His work with code-breaking also made him a pioneer in

basic concepts of probability.)

(Al-Kindi, calledThe Arab Philosopher, can not be

considered among the greatest of mathematicians,

but was one of the most influential general

scientists between Aristotle and da Vinci.)

He appears on Cardano's List of 12 Greatest Geniuses.

(Al-Khowârizmi and Jabir ibn Aflah are the other

Islamic scientists on that list.)

__Al-Sabi Thabit ibn Qurra__ al-Harrani

(836-901) Harran, Iraq

Thabit produced important books in philosophy

(including perhaps the famous mystic workDe Imaginibus)

medicine, mechanics, astronomy, and especially several mathematical fields:

analysis, non-Euclidean geometry, trigonometry, arithmetic,

number theory.

As well as being an original thinker, Thabit was a key

translator of ancient Greek writings; he translated

Archimedes' otherwise-lostBook of Lemmasand applied

one of its methods to construct a regular heptagon.

He developed an important new cosmology superior

to Ptolemy's (and which, though it was not heliocentric,

may have inspired Copernicus).

He was perhaps the first great mathematician to take the

important step of emphasizing

real numbers rather than either rational numbers or geometric sizes.

He worked in plane and spherical trigonometry,

and with cubic equations.

He was an early practitioner of calculus and seems to

have been first to take the integral of√x.

Like Archimedes, he was

able to calculate the area of an ellipse,

and to calculate the volume of a paraboloid.

He produced an elegant generalization of the

Pythagorean Theorem:

AC^{ 2}+ BC^{ 2}= AB (AR + BS)

(Here the triangle ABC is not a right triangle, but R and S are located

on AB to give the equal angles ACB = ARC = BSC.)

Thabit also worked in number theory where he is especially famous

for his theorem about amicable numbers.

(Thabit ibn Qurra's Theorem was rediscovered by Fermat and Descartes, and later

generalized by Euler.)

While many of his discoveries in geometry, plane and spherical trigonometry,

and analysis (parabola quadrature, trigonometric law, principle

of lever) duplicated work by Archimedes and Pappus, Thabit's

list of novel achievements is impressive.

Among the several great and famous Baghdad geometers,

Thabit may have had the greatest genius.

__Ibrahim ibn Sinan__ ibn Thabit ibn Qurra

(908-946) Iraq

Ibn Sinan, grandson of Thabit ibn Qurra, was one of the greatest

Islamic mathematicians and might have surpassed his famous grandfather

had he not died at a young age.

He was an early pioneer of analytic geometry,

advancing the theory of integration, applying algebra to synthetic geometry,

and writing on the construction of conic sections.

He produced a new proof of Archimedes' famous formula for the area of

a parabolic section.

He worked on the theory of area-preserving transformations, with

applications to map-making.

He also advanced astronomical theory, and wrote a treatise on sundials.

__Mohammed ibn al-Hasn (Alhazen) `Abu Ali' ibn al-Haytham__ al-Basra

(965-1039) Iraq, Egypt

Al-Hassan ibn al-Haytham (Alhazen)

made contributions to math, optics, and astronomy

which eventually influenced Roger Bacon, Regiomontanus, da Vinci, Copernicus,

Kepler, Galileo, Huygens, Descartes and Wallis,

thus affecting Europe's Scientific Revolution.

While Aristotle thought vision arose from rays sent from the eye

to the viewed object, and Ptolemy thought light rays originated

from objects, Alhazen understand that an object's light is

reflected sunlight.

He's been called the best scientist of the Middle Ages;

hansBook of Opticshas been called the most important

physics text prior to Newton;

his writings in physics anticipate the Principle of Least Action,

Newton's First Law of Motion, and the notion that white light

is composed of the color spectrum.

(Like Newton, he favored a particle theory of light over the

wave theory of Aristotle.)

His other achievements in optics include improved lens design,

an analysis of the camera obscura, Snell's Law,

an early explanation for the rainbow,

a correct deduction from refraction of atmospheric thickness,

and experiments on visual perception.

He studied optical illusions and was first to explain

psychologically why the Moon appears to be larger when near

the horizon.

He also did work in human anatomy and medicine.

(In a famous leap of over-confidence he claimed he could

control the Nile River; when the Caliph ordered him to do so,

he then had to feign madness!)

Alhazen has been called the "Father of Modern Optics,"

the "Founder of Experimental Psychology" (mainly for his work with

optical illusions), and,

because he emphasized hypotheses and experiments,

"The First Scientist."In number theory, Alhazen worked with perfect numbers, Mersenne primes,

and the Chinese Remainder Theorem.

He stated Wilson's Theorem (which is sometimes called Al-Haytham's Theorem).

Alhazen introduced the Power Series Theorem

(later attributed to Jacob Bernoulli).

His best mathematical work was with plane and solid geometry,

especially conic sections; he calculated the areas of lunes, volumes

of paraboloids, and constructed a regular heptagon using

intersecting parabolas.

He solved Alhazen's Billiard Problem (originally posed as a problem

in mirror design), a difficult construction which continued to intrigue

several great mathematicians including Huygens.

To solve it, Alhazen needed to

anticipate Descartes' analytic geometry,

anticipate Bézout's Theorem,

tackle quartic equations

and develop a rudimentary integral calculus.

Alhazen's attempts to prove the Parallel Postulate make him (along

with Thabit ibn Qurra) one of the earliest

mathematicians to investigate non-Euclidean geometry.

__Abu al-Rayhan Mohammed ibn Ahmad al-Biruni__

(973-1048) Khorasan (Uzbekistan)

Al-Biruni (Alberuni) was an extremely outstanding scholar,

far ahead of his time, sometimes shown with

Alkindus and Alhazen as one of the greatest Islamic polymaths,

and sometimes compared to Leonardo da Vinci.

He is less famous in part because he lived in a remote part of the

Islamic empire.

He was a great linguist; studied the original works of Greeks and Hindus;

is famous for debates with his contemporary Avicenna;

studied history, biology, mineralogy,

philosophy, sociology, medicine and more;

is called the Father of Geodesy and

the Father of Arabic Pharmacy;

and was one of the greatest astronomers.

He was an early advocate of the Scientific Method.

He was also noted for his poetry.

He invented (but didn't build) a geared-astrolabe clock,

and worked with springs and hydrostatics.

He wrote prodigiously on all scientific topics (his writings are estimated

to total 13,000 folios); he was especially noted for

his comprehensive encyclopedia about India, andShadows,

which starts from notions about shadows but develops much astronomy

and mathematics.

He anticipated future advances

including Darwin's natural selection,

Newton's Second Law, the immutability of elements,

the nature of the Milky Way, and much modern geology.

Among several novel achievements in astronomy, he used

observations of lunar eclipse to deduce relative longitude,

estimated Earth's radius most accurately,

believed the Earth rotated on its

axis and may have accepted heliocentrism as a possibility.

In mathematics, he was first to apply the Law of Sines

to astronomy, geodesy, and cartography;

anticipated the notion of polar coordinates;

invented the azimuthal equidistant map projection in common use today,

as well as a polyconic method

now called the Nicolosi Globular Projection;

found trigonometric solutions to polynomial equations;

did geometric constructions including angle trisection;

and wrote on arithmetic, algebra, and combinatorics

as well as plane and spherical trigonometry and geometry.

(Al-Biruni's contemporary Avicenna was not particularly a mathematician

but deserves mention as an advancing scientist, as does Avicenna's disciple

Abu'l-Barakat al-Baghdada, who lived about a century later.)Al-Biruni has left us what seems to be the oldest surviving mention

of the Broken Chord Theorem (if M is the midpoint of circular arc ABMC,

and T the midpoint of "broken chord" ABC, then MT is perpendicular to BC).

Although he himself attributed the theorem to Archimedes,

Al-Biruni provided several novel proofs for, and useful corollaries of,

this famous geometric gem.

While Al-Biruni may lack the influence and mathematical brilliance to

qualify for the Top 100, he deserves recognition as one of the

greatest applied mathematicians before the modern era.

__Omar al-Khayyám__

Omar Khayyám (aka Ghiyas od-Din Abol-Fath Omar ibn Ebrahim

Khayyam Neyshaburi) was one of the greatest Islamic mathematicians.

He did clever work with geometry, developing

an alternate to Euclid's Parallel Postulate and then deriving the

parallel result using theorems

basé surKhayyam-Saccheri quadrilateral.

He derived solutions to cubic equations using the intersection

of conic sections with circles.

Remarkably, he stated that the cubic solution could not be achieved

with straightedge and compass, a fact that wouldn't

be proved until the 19th century.

Khayyám did even more important work in algebra, writing

an influential textbook, and developing new solutions for

various higher-degree equations.

He may have been first to develop Pascal's Triangle (which is still

called Khayyám's Triangle in Persia), along with the

essential Binomial Theorem (Al-Khayyám's Formula):

(x+y)^{n}= n! Σ x^{k}y^{n-k}/ k!(n-k)!Khayyám was also an important astronomer; he measured the

year far more accurately than ever before,

improved the Persian calendar, built a famous star map,

and believed that the Earth rotates on its axis.

He was a polymath: in addition to being a philosopher of far-ranging scope,

he also wrote treatises on music, mechanics and natural science.

He was noted for deriving his theories from science rather than religion.

Despite his great achievements in algebra, geometry,

astronomy, and philosophy, today Omar al-Khayyám is most

famous for his rich poetry (The Rubaiyat of Omar Khayyám).

Bháscara II (also called Bhaskaracharya)

may have been the greatest of the Hindu

mathematicians. He made achievements in several fields

of mathematics including some Europe wouldn't learn until the time of Euler.

His textbooks dealt with many matters, including solid geometry, combinations,

and advanced arithmetic methods.

He was also an astronomer.

(It is sometimes claimed that his

equations for planetary motions anticipated the Laws of Motion

discovered by Kepler and Newton, but this claim is doubtful.)

In algebra, he solved various equations including 2nd-order

Diophantine, quartic, Brouncker's and Pell's equations.

HisChakravala method, an early application of mathematical induction

to solve 2nd-order equations, has been called "the finest thing

achieved in the theory of numbers before Lagrange"

(although a similar statement was made about one of Fibonacci's theorems).

(Earlier Hindus, including Brahmagupta, contributed to this method.)

In several ways he anticipated calculus: he used Rolle's Theorem;

he may have been first to use the fact

queré;sinx =cosx · dx

and he once wrote that multiplication by0/0could be "useful

in astronomy."

In trigonometry, which he valued for its own beauty as well

as practical applications, he developed spherical trig and was first to

present the identity

sina+b

=sina ·cosb

+sinb ·cosunBháscara's achievements came centuries before similar

discoveries in Europe.

It is an open riddle of history whether any of Bháscara's

teachings trickled into Europe in time to influence its

Scientific Renaissance.

**Leonardo `Bigollo'** Pisano (Fibonacci)

Leonardo (known today as Fibonacci)

introduced the decimal system and other new methods of arithmetic to Europe,

and relayed the mathematics of the Hindus, Persians, and Arabs.

Others, especially Gherard of Cremona, had translated Islamic

mathematics, e.g. the works of al-Khowârizmi, into Latin,

but Leonardo was the influential teacher.

(Two centuries earlier, the mathematician-Pope, Gerbert of Aurillac,

had tried unsuccessfully to introduce the decimal system to Europe.)

Leonardo also re-introduced older Greek ideas like

Mersenne numbers and Diophantine equations.

His writings cover a very broad range including

new theorems of geometry,

methods to construct and convert Egyptian fractions

(which were still in wide use),

irrational numbers,

the Chinese Remainder Theorem,

theorems about Pythagorean triplets,

and the series 1, 1, 2, 3, 5, 8, 13, ….

which is now linked with the name Fibonacci.

In addition to his great historic importance and fame

(he was a favorite of Emperor Frederick II),

Leonardo `Fibonacci' is called "the greatest number theorist

between Diophantus and Fermat" and "the mosttalent

mathematician of the Middle Ages."Leonardo is most famous for his book

Liber Abaci, but

hansLiber Quadratorumprovides the best demonstration of his skill.

He definedcongruumsand proved theorems about them,

including a theorem establishing the conditions for three square numbers

to be in consecutive arithmetic series; this has been called

the finest work in number theory prior to Fermat

(although a similar statement was made about one of Bhaskara II's theorems).

Although often overlooked, this work includes a proof

den = 4case of Fermat's Last Theorem.

(Leonardo's proof of FLT4 is widely ignored or considered incomplete.

I'm preparing a page to consider that question.

Al-Farisi was another ancient mathematician who noted FLT4, although

attempting no proof.)

Another of Leonardo's noteworthy achievements was proving that

the roots of a certain cubic equation could not have any of the

constructible forms Euclid had outlined in Book 10 of hisElements.

He also wrote on, but didn't prove, Wilson's Theorem.Leonardo provided Europe with the decimal system,

algebra and the 'lattice' method of multiplication, all

far superior to the methods then in use.

He introduced notation like3/5; his clever extension

of this for quantities like5 yards, 2 feet, and 3 inches

is more efficient than today's notation.

It seems hard to believe but before the decimal

system, mathematicians had no notation for zero.

Referring to this system,

Gauss was later to exclaim "To what heights would science

now be raised if Archimedes had made that discovery!"Some histories describe him as bringing Islamic mathematics

to Europe, but in Fibonacci's own preface toLiber Abaci,

he specifically credits the Hindus:… as a consequence of marvelous instruction in the art,

to the nine digits of the Hindus, the knowledge of the art

very much appealed to me before all others, and for it I

realized that all its aspects were studied in Egypt, Syria,

Greece, Sicily, and Provence, with their varying methods;

… But all this even, and the algorism, as well as the art

of Pythagoras, I considered as almost a mistake in respect

to the method of the Hindus. Derfor,

embracing more stringently that method of the Hindus, and

taking stricter pains in its study, while adding certain

things from my own understanding and inserting also certain

things from the niceties of Euclid's geometric art, I have

striven to compose this book in its entirety as understandably

as I could, …Had the Scientific Renaissance begun in the Islamic Empire,

someone like al-Khowârizmi would have greater historic

significance than Fibonacci,

but the Renaissance did happen in Europe.

Liber Abaci's summary of the

decimal system has been called "the most important sentence

ever written."

Even granting this to be an exaggeration, there is no doubt

that the Scientific Revolution owes a huge debt to

Leonardo `Fibonacci' Pisano.

__Abu Jafar Muhammad Nasir al-Din al-Tusi__

Al-Tusi was one of the greatest Islamic polymaths, working

in theology, ethics, logic, astronomy, and other fields of science.

He was a famous scholar and prolific writer,

describing evolution of species, stating that the Milky Way was

composed of stars,

and mentioning conservation of mass in his writings on chemistry.

He made a wide range of contributions to astronomy, and (along

with Omar Khayyám) was one of the

most significant astronomers between Ptolemy and Copernicus.

He improved on the Ptolemaic model of planetary orbits, and even wrote

about (though rejecting) the possibility of heliocentrism.

(Allegedly one of his students hypothesized elliptical orbits.)Tusi is most famous for his mathematics.

He advanced algebra, arithmetic, geometry, trigonometry, and even foundations,

working with real numbers and lengths of curves.

For his texts and theorems, he may be called the "Father of Trigonometry;"

he was first to properly state and prove several theorems

of planar and spherical trigonometry including the Law of Sines,

and the (spherical) Law of Tangents.

He wrote important

commentaries on works of earlier Greek and Islamic mathematicians;

he attempted to prove Euclid's Parallel Postulate.

Tusi's writings influenced European mathematicians including Wallis;

his revisions of the Ptolemaic model led him to the Tusi-couple,

a special case of trochoids usually called Copernicus' Theorem,

though historians have concluded Copernicus

discovered this theorem by reading Tusi.

There were several important Chinese mathematicians

in the 13th century, of whom Qin Jiushao (Ch'in Chiu-Shao) may have had

particularly outstanding breadth and genius.

Qin's textbook discusses various algebraic procedures,

includes word problems requiring quartic or quintic equations,

explains a version of Horner's Method for finding solutions

to such equations,

includes Heron's Formula for a triangle's area, and

introduces the zero symbol and decimal fractions.

Qin's work on the Chinese Remainder Theorem was very impressive,

finding solutions in cases which later stumped Euler.Other great Chinese mathematicians of that era are

Li Zhi, Yang Hui (Pascal's Triangle is still

called Yang Hui's Triangle in China), and Zhu Shiejie.

Their teachings did not make their way to Europe, but were read by the

Japanese mathematician Seki, and possibly by Islamic

mathematicians like Al-Kashi.

Although Qin was a soldier and governor noted for corruption, with

mathematics just a hobby, I've chosen him to represent this

group because of the key advances which appear first in his writings.

Zhu Shiejie (Chu Shih-Chieh) was more famous and influential

than Qin; historian George Sarton called him

"one of the greatest mathematicians … of all time."

His bookJade Mirror of the Four Unknownsstuderte

multivariate polynomials and is considered

the best mathematics in ancient China and describes methods not

rediscovered for centuries; par exemple

Zhu anticipated the Sylvester matrix method for solving simultaneous

polynomial equations.

__Kamal al-Din Abu'l Hasan Muhammad Al-Farisi__

Al-Farisi was a student of al-Shirazi who in turn was

a student of al-Tusi. He and al-Shirazi are especially noted for the

first correct explanation of the rainbow.

Al-Farisi made several other corrections in his comprehensive

commentary on Alhazen's textbook on optics.Al-Farisi made several contributions to number theory. He improved

Thabit's theorem about amicable numbers, made important new observations

about the binomial coefficients, and noted the N=4 case of Fermat's Last

Theorem.

In addition to his work with amicable numbers, he is

especially noted for his improved proof of

Euclid's Fundamental Theorem of Arithmetic.

__Levi ben Gerson `Gersonides'__

Gersonides (aka Leo de Bagnols, aka RaLBaG) was a Jewish scholar of

great renown, preferring science and reason over religious orthodoxy.

He wrote important commentaries on Aristotle, Euclid, the Talmud,

and the Bible; he is most famous for his

livreMilHamot Adonai("The Wars of the Lord") which

touches on many theological questions.

He was likely the most talented scientist of his time:

he invented the "Jacob's Staff" which became an important navigation

tool; described the principles of the camera obscura; etc.

In mathematics, Gersonides wrote texts on trigonometry,

calculation of cube roots, rules of arithmetic, etc.;

and gave rigorous derivations of rules of combinatorics.

He was first to make explicit use of mathematical induction.

At that time, "harmonic numbers" referred to integers with only 2 and

3 as prime factors; Gersonides solved a problem of music theory with

an ingenious proof that there were no consecutive harmonic numbers

larger than (8,9).

(In fact, (2^3,3^2) is the only pair of consecutive powers;

this is Catalan's Conjecture and was first proved in the 21st century.)

Levi ben Gerson published only in Hebrew so,

although some of his work was translated into

Latin during his lifetime, his influence was limited; much of his work was re-invented

three centuries later; and many histories of math overlook him altogether.Gersonides was also an outstanding astronomer. He proved that

the fixed stars were at a huge distance, and found other

flaws in the Ptolemaic model.

But he specifically rejected heliocentrism, noteworthy since it

implies that heliocentrism was under consideration at the time.

__Nicole Oresme__

Oresme was of lowly birth but excelled at school,

became a young professor, and soon personal chaplain to King Charles V.

(One of his early teachers was Jean Buridan (ca 1300-ca 1358), famous

for positing what became Newton's First Law of Motion, contrary to

Aristotelian dogma.)

The King commissioned him to translate the works of

Aristotle into French (with Oresme thus playing key roles in

the development of both French science and French language), and

rewarded him by making him a Bishop.

He wrote several books; was a renowned philosopher and natural scientist

(challenging several of Aristotle's ideas);

contributed to economics (e.g. anticipating Gresham's Law)

and to optics (he was first to posit curved refraction).

Although the Earth's annual orbit around the Sun was left to Copernicus,

Oresme was among the pre-Copernican thinkers to claim clearly that the

Earth spun daily on its axis.In mathematics, Oresme observed that the integers were

equinumerous with the odd integers; was first to use fractional

(and even irrational) exponents;

introduced the symbolfor addition;+

was first to write about general curvature;

and, most famously, first to prove the divergence of the harmonic series.

Oresme used a graphical diagram to demonstrate the Merton College Theorem

(a discovery related to Galileo's Law of Falling Bodies

made by Thomas Bradwardine, et al); it is said this was the

first abstract graph.

(Some believe that this effort inspired

Descartes' coordinate geometry and Galileo.)

Oresme was aware of Gersonides' work on harmonic numbers and was among

those who attempted to link music theory to the ratios of celestial

orbits, writing "the heavens are like a man who sings a melody and

at the same time dances, thus making music … in song and in action."

Oresme's work was influential; with several discoveries ahead of

his time, Oresme deserves to be better known.

Madhava, also known as Irinjaatappilly Madhavan Namboodiri,

founded the important Kerala school of mathematics and astronomy.

If everything credited to him was his own work, he was a

truly great mathematician.

His analytic geometry preceded and surpassed Descartes',

and included differentiation and integration.

Madhava also did work with continued fractions, trigonometry,

and geometry.

He has been called the "Founder of Mathematical Analysis."

Madhava is most famous for his work with Taylor series,

discovering identities like

,sinq = q – q^{3}/3! +

q^{5}/5! – …

formulae for, including the one attributed to Leibniz,π

and the then-best known approximation

π ≈ 104348 / 33215.Despite the accomplishments of the Kerala school, Madhava probably does not

deserve a place on our List. There were several other great mathematicians

who contributed to Kerala's achievements, some of which were made 150 years

after Madhava's death.

More importantly, the work was not propagated outside Kerala,

so had almost no effect on the development of mathematics.

__Ghiyath al-Din Jamshid Mas'ud Al-Kashi__

(ca 1380-1429) Iran, Transoxania (Uzbekistan)

Al-Kashi was among the greatest calculators in the

ancient world; wrote important texts applying arithmetic and algebra

to problems in astronomy, mensuration and accounting;

and developed trig tables far more accurate than earlier tables.

He worked with binomial coefficients,

invented astronomical calculating machines, developed spherical trig,

and is credited with various theorems of trigonometry

including the Law of Cosines, which is sometimes called Al-Kashi's Theorem.

He is sometimes credited with the invention of decimal fractions

(though he worked mainly with sexagesimal fractions),

and a method like Horner's to calculate roots.

However decimal fractions had been used earlier, e.g. by Qin Jiushao;

and Al-Kashi's root calculations may also have been

derived from Chinese texts by Qin Jiushao or Zhu Shiejie.Using his methods, al-Kashi calculated

πriktig

to 17 significant digits, breaking Madhava's record.

(This record was subsequently broken by

relative unknowns: a German ca. 1600, John Machin 1706.

In 1949 thecalculation record was held brieflyπ

by John von Neumann and the ENIAC.)

__Nicholas Kryffs of Cusa__

Nicholas Kryffs (aka Nikolaus Cusanus) was an

astronomer and ordained priest. His main work was in philosophy

(he wrote "All we know of the truth is that the absolute truth,

such as it is, is beyond our reach") and theology; but he

also made proposals for scientific experimentation, e.g.

the use of water-clocks to investigate the speed of falling

bodies, and the use of weighing balances to quantify the transfer

of mass from soil to growing plant.

Nicholas was also a mathematician: he wrote on geometric problems,

calendar reform,

and invented theBorda Countmethod of balloting (although

it is named after its 18th-century rediscoverer),

but his work doesn't qualify him for our list.

He belongs in our story because he deduced that the Earth

orbited the Sun before Copernicus did, and made a deduction that Copernicus

missed: that the universe is immensely huge or infinite; and the fixed stars

may have their own planets with their own life forms.

He was far ahead of his time, but his writings eventually influenced

Galileo, Leibniz, and another mathematician-priest more famous than himself:

Giordano Bruno, who wrote

"If Nicholas (of Cusa) had not been hindered by his priest's vestment,

he would have even been greater than Pythagoras!"

__Johannes Müller von Königsberg `Regiomontanus'__

Regiomontanus was a prodigy who entered University at age eleven,

studied under the influential Georg von Peuerbach,

and eventually collaborated with him.

He was an important astronomer;

he found flaws in Ptolemy's system (thus influencing Copernicus),

realized lunar observations could be used to determine

longitude, and may have believed in heliocentrism.

His ephemeris was used by Columbus, when shipwrecked on Jamaica,

to predict a lunar eclipse,

thus dazzling the natives and perhaps saving his crew.

More importantly, Regiomontanus was one of the most influential mathematicians

of the Middle Ages; he published trigonometry textbooks and tables,

as well as the best textbook on arithmetic and algebra of his time.

(Regiomontanus lived shortly after Gutenberg,

and founded the first scientific press.)

He was a prodigious reader of Greek and Latin translations,

and most of his results were copied from Greek works

(or indirectly from Arabic writers, especially Jabir ibn Aflah);

however he improved or reconstructed many of the proofs, and

often presented solutions in both geometric and algebraic form.

His algebra was more symbolic and general than his predecessors';

he solved cubic equations (though not the general case);

applied Chinese remainder methods, and worked in number theory.

He posed and solved a variety of clever geometric puzzles, including

his famous angle maximization problem.

Regiomontanus was also an instrument maker, astrologer, and Catholic bishop.

He died in Rome where he had been called to

advise the Pope on the calendar; his early death may have delayed

the needed reform until the time of Pope Gregory.

Leonardo da Vinci is most renowned for his paintings —

Mona Lisa

etThe Last Supperare among the most discussed

and admired paintings ever —

but he did much other work and was probably the most

talented, versatile and prolific polymath ever to live;

his writings exceed 13,000 folios.

He developed new techniques, and principles of perspective geometry,

for drawing, painting and sculpture;

he was also an expert architect and engineer;

and surely the most prolific inventor of all time.

Although most of his paper designs were never built, Leonardo's inventions

include reflecting and refracting telescope, adding machine,

parabolic compass,

improved anemometer, parachute, helicopter, flying ornithopter,

several war machines (multi-barreled gun, steam-driven cannon,

tank, giant crossbow, finned mortar shells, portable bridge),

pumps, an accurate spring-operated clock,

bobbin winder, robots, scuba gear, an elaborate musical

instrument he called the 'viola organista,' and more.

(Some of his designs, including the viola organista, his

parachute, and a large single-span bridge, were finally built five

centuries later; and worked as intended.)

Like another genius (Albert Einstein) da Vinci was intrigued

by the science of river meanders; there is a meandering river

in the background ofMona Lisa.

His scientific writings are much more extensive and thorough

than is generally appreciated; he made advances in anatomy, botany,

and many other fields of science;

he developed much mechanics including the theory of the arch;

he developed an octant-based map projection;

he was first to conceive of plate tectonics; and,

in one cryptic reference, shows belief in heliocentrism.

He was also a poet and musician.He had little formal training in mathematics until he was

in his mid-40's, when he and Luca Pacioli (the other great Italian

mathematician of that era) began tutoring each other.

Despite this slow start, he did make novel achievements in mathematics:

he was first to note the simple classification of symmetry

groups on the plane, achieved interesting bisections

and mensurations, advanced the craft of descriptive geometry,

and developed an approximate solution

to the circle-squaring problem.

He was first to discover the 60-vertex shape now called "buckyball."

(Leonardo is also widely credited with the elegant two-hexagon proof

of the Pythagorean Theorem, but this authorship

appears to be a myth.)

Along with Archimedes, Alberuni, Leibniz, and J. W. von Goethe,

Leonardo da Vinci was among the greatest geniuses ever;

but none of these appears on Hart's List

of the Most Influential Persons in History:

genius doesn't imply influence.

(However, M.I.T.'s Pantheon project, using the statistics of

on-line biographies, prepared a list of the Thirty-Five (or Eighty)

Most Influential Persons in History; in addition to five (six) names already

on our list and Hart's — Aristotle, Newton, Einstein, Galileo, Euclid (and Descartes) —

their list includes four (seven) other mathematicians

missing from Hart's list: Plato, Leonardo, Pythagoras, Archimedes

(and Thales, Pascal, Ptolemy).Leonardo was also a writer and philosopher. Among his notable adages are

"Simplicity is the ultimate sophistication,"

and "The noblest pleasure is the joy of understanding,"

and "Human ingenuity … will never discover any inventions more

beautiful, more simple or more practical than those of nature."

__Nicolaus Copernicus__

The European Renaissance developed in 15th-century

Italy, with the blossoming of great art, and

as scholars read books by great Islamic scientists like Alhazen.

The earliest of these great Italian polymaths were largely not noted

for mathematics, and Leonardo da Vinci began serious math study

only very late in life, so the best candidates for mathematical greatness

in the Italian Renaissance were foreigners.

Along with Regiomontanus from Bavaria, there was an even more famous

man from Poland.Nicolaus Copernicus (Mikolaj Kopernik) was a polymath:

he studied law and medicine;

published poetry; contemplated astronomy;

worked professionally as a church scholar and diplomat;

and was also a painter.

He studied Islamic works on astronomy and geometry

at the University of Bologna, and eventually wrote a book of great impact.

Although his only famous theorem of mathematics (that certain trochoids are

straight lines) may have been derived from Oresme's work,

or copied from Nasir al-Tusi,

it was mathematical thought that led Copernicus to the conclusion

that the Earth rotates around the Sun.

Despite opposition from the Roman church, this discovery

led, via Galileo, Kepler and Newton, to the Scientific Revolution.

For this revolution, Copernicus is ranked #19 on Hart's List

of the Most Influential Persons in History; however I think

there are several reasons why Copernicus' importance may be exaggerated:

(1) Copernicus' system still used circles and epicycles, so it

was left to Kepler to discover the facts of elliptical orbits;

(2) he retained the notion of a sphere of fixed stars,

thus missing the unifying insight that our sun is one of many;

(3) Giordano Bruno (1548-1600), who built on Copernicus'

discovery, was a better and more influential scientist,

anticipating some of Galileo's concepts;

and (4) the Scientific Revolution didn't really get underway

until the invention of the telescope, which would have soon

led to the discovery of heliocentrism in any event.Until the Protestant Reformation, which began about the time

of Copernicus' discovery, European scientists were reluctant to challenge

the Catholic Church and its belief in geocentrism.

Copernicus' book was published only posthumously.

It remains controversial whether earlier Islamic or Hindu mathematicians

(or even Archimedes with hisThe Sand Reckoner)

believed in heliocentrism, but

were also inhibited by religious orthodoxy.

__Girolamo Cardano__

Girolamo Cardano (or Jerome Cardan) was a highly

respected physician and was first to describe typhoid fever.

He was also an accomplished gambler and

chess player and wrote an early book on probability.

He was also a remarkable inventor:

the combination lock, an advanced gimbal, a

ciphering tool, and the Cardan shaft with universal joints are all

his inventions and are in use to this day.

(The U-joint is sometimes called the Cardan joint.)

He also helped develop the camera obscura.

Cardano made contributions to physics: he noted that

projectile trajectories are parabolas, and may have been first

to note the impossibility of perpetual motion machines.

He did work in philosophy, geology, hydrodynamics,

music; he wrote books on medicine and

an encyclopedia of natural science.But Cardano is most remembered for his achievements in mathematics.

He was first to publish general solutions

to cubic and quartic equations,

and first to publish the use of complex numbers in calculations.

(Cardano's Italian colleagues deserve much credit:

Ferrari first solved the quartic, he or Tartaglia the cubic; et

Bombelli first treated the complex numbers as numbers in their own right.

Cardano may have been thederniergreat mathematician

unwilling to deal with negative numbers: his treatment of cubic

equations had to deal withax^{3}– bx + c = 0

etaxqui^{3}– bx = c

two different cases.)

Cardano introduced binomial coefficients and the Binomial

Theorem, and introduced and solved the geometric

hypocycloid problem, as well as other geometric theorems

(e.g. the theorem underlying the 2:1 spur wheel

which converts circular to reciprocal rectilinear motion).

Cardano is credited withCardano's Ring Puzzle,

still manufactured today and related to the

Tower of Hanoipuzzle.

(This puzzle may predate Cardano, and may even have been

known in ancient China.)

Da Vinci and Galileo may have been more influential

than Cardano, but of the three great generalists

in the century before Kepler, it seems clear that Cardano was the

most accomplished mathematician.Cardano's life had tragic elements.

Throughout his life he was tormented that his

father (a friend of Leonardo da Vinci) married

his mother only after Cardano was born.

(And his mother tried several times to abort him.)

Cardano's reputation for gambling and aggression interfered

with his career.

He practiced astrology and was imprisoned for heresy

when he cast a horoscope for Jesus.

(This and other problems were due in part to

revenge by Tartaglia for Cardano's

revealing his secret algebra formulae.)

His son apparently murdered his own wife.

Leibniz wrote of Cardano:

"Cardano was a great man with all his faults;

without them, he would have been incomparable."

__Rafael Bombelli__

Bombelli was a talented engineer who wrote

an algebra textbook sometimes considered

one of the foremost achievements of the 16th century.

Although incorporating work by Cardano, Diophantus and

possibly Omar al-Khayyám,

the textbook was highly original and extremely influential.

Leibniz and Huygens were among many who praised his work.

Although noted for his new ideas of arithmetic, Bombelli based

much of his work on geometric ideas, and even pursued complex-number

arithmetic to an angle-trisection method.

He was the first European of his era to use continued fractions,

using them for a new square-root procedure.

In his textbook he also introduced new symbolic notations,

allowed negative and complex numbers, and gave the rules for

manipulating these new kinds of numbers.

Bombelli is often called the Inventor of Complex Numbers.

__François Viète__

François Viète (or Franciscus Vieta) was a French

nobleman and lawyer who was a favorite of King Henry IV and eventually

became a royal privy councillor.

In one notable accomplishment he broke the Spanish diplomatic code,

allowing the French government to read Spain's messages and

publish a secret Spanish letter; this apparently led to the end

of the Huguenot Wars of Religion.More importantly, Vieta was certainly

the best French mathematician prior to Descartes and Fermat.

He laid the groundwork for modern mathematics;

his works were the primary teaching for both Descartes and Fermat;

Isaac Newton also studied Vieta.

In his role as a young tutor Vieta used decimal numbers before they

were popularized by Simon Stevin and may have guessed

that planetary orbits were ellipses before Kepler.

Vieta did work in geometry, reconstructing and publishing proofs

for Apollonius' lost theorems, including all ten cases of the

generellProblem of Apollonius.

Vieta also used his new algebraic techniques

to construct a regular heptagon.

He discovered several trigonometric identities including

a generalization of Ptolemy's Formula,

the latter (then calledprosthaphaeresis)

providing a calculation shortcut similar to logarithms in that

multiplication is reduced to

addition (or exponentiation reduced to multiplication).

Vieta also used trigonometry to find real solutions

to cubic equations for which the Italian methods had

required complex-number arithmetic; he also used trigonometry

to solve a particular 45th-degree equation that had been

posed as a challenge.

Such trigonometric formulae revolutionized calculations and may

even have helped stimulate the development and use of

logarithms by Napier and Kepler.

He developed the first infinite-product formula forπ.

In addition to his geometry and trigonometry, he also

found results in number theory, but

Vieta is most famous for his systematic use of decimal notation

and variable letters, for which he is sometimes called the "Father of

Modern Algebra."

(Vieta used A,E,I,O,U for unknowns and consonants for parameters;

it was Descartes who first used X,Y,Z for unknowns and A,B,C for parameters.)

In his works Vieta emphasized the relationships between algebraic

expressions and geometric constructions.

One key insight he had is that addends must be homogeneous

(i.e., "apples shouldn't be added to oranges"), a seemingly trivial idea

but which can aid intuition even today.Descartes, who once wrote "I began where Vieta finished," is

now extremely famous, while Vieta is much less known. (He isn't

even mentioned oncein Bell's famousMen of Mathematics.)

Many would now agree this is due in large measure to Descartes'

deliberate deprecations of competitors in his quest for personal glory.

(Vieta wasn't particularly humble either, calling himself

the "French Apollonius.")

PI := 2 Y := 0 LOOP: Y := SQRT(Y + 2) PI := PI * 2 / Y IF (more precision needed) GOTO LOOP

Vieta's formula for

πis clumsy to express

without trigonometry, even with modern notation.

Easiest may be to consider it the result of the BASIC program above.

Using this formula, Vieta constructed an approximation toπ

that was best-yet by a European, though not as accurate as al-Kashi's

two centuries earlier.

__Giordano Bruno__

Bruno wrote on mnemonics, philosophy, cosmology

and more, and further developed Nicholas of Cusa's

pandeism(though it wasn't then known by that name).

He embraced not only Copernicus' notion that the Earth rotates around

the Sun, but thought the Sun itself moved and that, as Nicholas

had conjectured, the universe was infinite and teemed with other worlds.

This is a conclusion the great Kepler didpasmake.

(Galileo's stance on this question is ambiguous: he knew the price

Bruno had paid for expressing this belief.)

Bruno was egotistical and obnoxious, and his teachings (most

especially his belief in multiple solar systems) infuriated the

Roman Church, which had him hung upside-down, spikes driven through his

mouth so he could not utter further heresies, and burned alive.

Before his execution he was tortured for seven years but never recanted,

instead saying "Maybe you who condemn me are in greater fear than I

who am condemned."One historian claims that it was Bruno's publications in 1584 and their

influence that defined the transition from medieval thought to modern

scientific thought.

William Gilbert (who laid the foundations of electricity and magnetism,

which is called the first important scientific discovery by an Englishman),

acknowledged the influence of Bruno's cosmology.

The Revolution started by Nicholas and Bruno was a revolt against

the Church's embracing Aristotle's teachings, as Bruno made clear when

he wrote "There is no absolute up or down, as Aristotle taught;

no absolute position in space…. Everywhere there is incessant relative

change … and the observer is always at the centre of things."

Bruno made few mathematical discoveries, if any; but some consider him to be

one of history's great geniuses, in a class with Johann von Goethe,

Leonardo da Vinci, Francis Bacon, Voltaire, Nietzsche, and some of the

very greatest mathematicians.

__Simon Stevin__

Stevin was one of the greatest practical scientists

of the Late Middle Ages.

He worked with Holland's dykes and windmills;

as a military engineer he developed fortifications and systems of flooding;

he invented a carriage with sails that traveled faster than with horses

and used it to entertain his patron, the Prince of Orange.

He discovered several laws of mechanics including those

for energy conservation and hydrostatic pressure.

He lived slightly before Galileo who is now much more famous,

but Stevin discovered the equal rate of falling bodies before Galileo did;

and his explanation of tides was better than Galileo's, though still incomplete.

He was first to write on the concept of unstable equilibrium.

He invented improved accounting methods, and (though also invented

at about the same time by Chinese mathematician Zhu Zaiyu

and anticipated by Galileo's father, Vincenzo Galilei)

the equal-temperament music scale.

He also did work in descriptive geometry,

trigonometry, optics, geography, and astronomy.In mathematics,

Stevin is best known for the notion of real numbers

(previously integers, rationals and irrationals were treated separately;

negative numbers and even zero and one were often not considered numbers).

He introduced (a clumsy form of) decimal fractions to Europe;

suggested a decimal metric

system, which was finally adopted 200 years later;

invented other basic notation like the symbol. √

Stevin proved several theorems about perspective geometry,

an important result in mechanics, and special cases of

the Intermediate Value Theorem later attributed to Bolzano and Cauchy.

Stevin's books, written in Dutch rather than Latin,

were widely read and hugely influential.

He was a very key figure in the development of modern European mathematics,

and may belong on our List.

__Jean Napier 8th of
Merchistoun__

Napier was a Scottish Laird who was a noted theologian

and thought by many to be a magician

(his nickname was Marvellous Merchiston).

Today, however, he is best known

for his work withlogarithms, a word he invented.

(Several others, including Archimedes, had anticipated the use of logarithms.)

He published the first large table

of logarithms and also helped popularize usage of the decimal point

and lattice multiplication.

He inventedNapier's Bones,

a crude hand calculator which could be used for division and

root extraction, as well as multiplication.

He also had inventions outside mathematics, especially

several different kinds of war machine.Napier's noted textbooks also contain an exposition

of spherical trigonometry.

Although he was certainly very clever (and had novel

mathematical insights not mentioned in this summary),

Napier proved no deep theorem

and may not belong in the Top 100.

Nevertheless, his revolutionary methods of arithmetic had

immense historical importance; his logarithm tables were used

by Johannes Kepler himself, and led to the Scientific Revolution.

Galileo discovered the laws of inertia (including rudimentary

forms of all three of Newton's laws of motion), falling bodies (including

parabolic trajectories), and the pendulum; he also introduced the notion

of relativity which later physicists found so fruitful.

(Although he admired Galileo greatly, Einstein's famous result

was somewhat misnamed: it depended on theabsoluteness,

not relativity, of the speed of light.)

Galileo discovered important principles of dynamics, including the

essential notion that the vector sum of forces produce an

acceleration.

(Aristotle seems not to have considered the notion of acceleration,

though his successor Strato of Lampsacus did write on it.)

Galileo may have been first to note that a larger body has less

relative cohesive strength than a smaller body.

He was a great inventor: in addition to being first to conceive

of a pendulum clock and of a thermometer, he developed

a new type of pump, the first compound-lens microscope,

and the best telescope, hydrostatic balance,

and cannon sector of his day.

As a famous astronomer, Galileo

pointed out that Jupiter's Moons, which he discovered, provide

a natural clock and allow a universal time to be determined

by telescope anywhere on Earth.

(This was of little use in ocean navigation since a ship's rocking

prevents the required delicate observations.

Galileo tried to measure the speed of light, but it was too fast for

him. However 66 years after Galileo discovered Jupiter's moons

and proposed using them as a clock,

the astronomer Roemer inferred the speed of light from that 'clock':

the clock had a discrepancy of up to seven minutes depending

on the Earth-Jupiter distance.)

Galileo's other astronomical discoveries also included sunspots

and lunar craters.Perhaps Galileo's most important astronomical discovery was the phases

of Venus. Ptolemy's epicycles, Copernicus' epicycles, and Kepler's ellipses

all gave almost the same solutions for planets' apparent positions,

but Ptolemy's system gave completely wrong predictions about

the phases of Venus.

Galileo's discovery of Venus' phases was the

critical fact which finally forced acceptance of heliocentrism.

(It's possible that Galileo's pursuit of Venus' phases was inspired in part

by Galileo's student, Benedetto Castelli.)

Just as modern inventors sometimes mail sealed envelopes to an

arbiter to establish precedence, Galileo sent an anagram to Kepler,

to later prove the date of his unpublished discovery.

(The anagram wasHaec immatura a me iam frustra leguntur o.y.qui

letters can be rearranged toCynthiae figuras aemulatur mater amorum.

These two Latin sentences translate respectively as

"I am now bringing these unripe things together in vain, Oy!" et

"The mother of love (Venus) copies the forms of Cynthia (the Moon).")Galileo's contributions outside physics and astronomy

were also enormous:

He made discoveries with the microscope he invented, and

made several important contributions to the early

development of biology.

Perhaps Galileo's most important contribution

was theDoctrine of Uniformity, the postulate that

c'estuniverselllaws of mechanics, in contrast to

Aristotelian and religious notions

of separate laws for heaven and earth.Galileo is often called the "Father of Modern Science"

because of his emphasis on experimentation.

His use of a ramp to discover his Law of Falling Bodies

was ingenious.

(For his experiments he started with a water-clock to measure time,

but found the beats reproduced by trained musicians to be more convenient.)

He understood that results needed to be repeated and averaged

(he minimized mean absolute-error for his curve-fitting

criterion, two centuries before Gauss and Legendre introduced

the mean squared-error criterion).

For his experimental methods and discoveries,

his laws of motion, and for (eventually) helping

to spread Copernicus' heliocentrism, Galileo may have been

the most influential scientist ever; han

ranks #12 on Hart's list of the Most Influential Persons in History.

(Despite these comments, it does appear that Galileo ignored

experimental results that conflicted with his theories.

For example, the Law of the Pendulum, based on Galileo's incorrect

belief that the tautochrone was the circle, conflicted

with his own observations.

Some of his other ideas were wrong; for example, he dismissed Kepler's

elliptical orbits and notion of gravitation and published a

very faulty explanation of tides.)

Despite his extreme importance to mathematical physics,

Galileo doesn't usually appear on

lists of greatestmathematicians.

However, Galileo did do work in pure mathematics;

he derived certain centroids and the parabolic shape

of trajectories using a rudimentary calculus,

and mentored Bonaventura Cavalieri, who extended Galileo's calculus;

he named (and may have been first to discover) the cycloid curve.

Moreover, Galileo was one of the first to write about

infinite equinumerosity (the "Hilbert's Hotel Paradox").

Galileo once wrote "Mathematics is the language in which God has written the

univers."

__Johannes Kepler__

Kepler was interested in astronomy from an early age,

studied to become a Lutheran minister,

became a professor of mathematics instead, then Tycho Brahe's understudy,

and, on Brahe's death, was appointed Imperial Mathematician at

the age of twenty-nine.

His observations of the planets with Brahe, along with his study

of Apollonius' 1800-year old work, led to Kepler's three Laws

of Planetary Motion, which in turn led directly to Newton's

Laws of Motion.

Beyond his discovery of these Laws (one of the most important

achievements in all of science), Kepler is also sometimes called

the "Founder of Modern Optics."

He furthered the theory of the camera obscura,

telescopes built fromdeuxconvex lenses,

and atmospheric refraction.

The question of human vision had been

considered by many great scientists including Aristotle, Euclid,

Ptolemy, Galen, Alkindus, Alhazen, and Leonardo da Vinci,

but it was Kepler who was first to explain the

operation of the human eye correctly and to note that retinal

images will be upside-down.

Kepler developed a rudimentary notion of universal gravitation,

and used it to produce the best explanation for tides before Newton;

however he seems not to have noticed that his empirical laws

implied inverse-square gravitation.

Kepler noticed Olbers' Paradox before Olbers' time and used it

to conclude that the Universe is finite.

Kepler ranks #75 on Michael Hart's famous list of

the Most Influential Persons in History. This rank, much lower than

that of Copernicus, Galileo or Newton, seems to me to underestimate Kepler's

importance, since it was Kepler's Laws, rather than just heliocentrism,

which were essential to the early development of mathematical physics.According to Kepler's Laws, the planets move at variable speed

along ellipses.

(Even Copernicus thought the orbits could be described with only circles.)

The Earth-bound observer is himself describing

such an orbit and in almost the same plane as the planets;

thus discovering the Laws would be a difficult challenge even for

someone armed with computers and modern mathematics.

(The very famous Kepler Equation relating a planet's

eccentric and anomaly is just one tool Kepler needed to develop.)

Kepler understood the importance of his remarkable discovery, even if

contemporaries like Galileo did not, writing:"I give myself up to divine ecstasy … My book is written.

It will be read either by my contemporaries or by posterity —

I care not which. It may well wait a hundred years for a

reader, as God has waited 6,000 years for someone to

understand His work."Kepler also once wrote "Mathematics is the archetype of the

beautiful."Besides the trigonometric results needed to discover his Laws,

Kepler made other contributions to mathematics.

He generalized Alhazen's Billiard Problem, developing the

notion of curvature.

He was first to notice that the set of Platonic regular solids

was incomplete if concave solids are admitted, and first

to prove that there were only 13Archimedean solids.

He proved theorems of solid geometry later discovered on

the famous palimpsest of Archimedes.

He rediscovered the Fibonacci series, applied it to

botany, and noted that the ratio of Fibonacci numbers

converges to theGolden Mean.

He was a key early pioneer in calculus,

and embraced the concept of continuity

(which others avoided due to Zeno's paradoxes);

his work was a direct inspiration for Cavalieri and others.

He developed the theory of logarithms and improved on

Napier's tables.

He developed mensuration methods and

anticipated Fermat's theorem on stationary points.

Kepler once had an opportunity to buy wine, which merchants measured

using a shortcut; with the famousKepler's Wine Barrel Problem,

he used his rudimentary calculus to deduce which barrel shape

would be the best bargain.Kepler reasoned that the structure of snowflakes was evidence

for the then-novel atomic theory of matter.

He noted that the obvious packing of cannonballs gave maximum

density (this became known asKepler's Conjecture; optimalitet

was proved among regular packings by Gauss, but it wasn't until

1998 that the possibility of denseruregelmessigpackings

was disproven).

In addition to his physics and mathematics,

Kepler wrote a science fiction novel,

and was an astrologer and mystic.

He had ideas similar to Pythagoras about numbers ruling the

cosmos (writing that the purpose of studying the world

"should be to discover the rational order and harmony which

has been imposed on it by God and which He revealed to us in the language of

mathematics").

Kepler's mystic beliefs even led to his own mother being

imprisoned for witchcraft.Johannes Kepler (along with Galileo, Fermat,

Huygens, Wallis, Vieta and Descartes)

is among the giants on whose shoulders Newton was proud to stand.

Some historians place him ahead of Galileo and Copernicus as the single most

important contributor to the early Scientific Revolution.

Chasles includes Kepler on a list of the six responsible for

conceiving and perfecting infinitesimal calculus (the other five are

Archimedes, Cavalieri, Fermat, Leibniz and Newton).

(www.keplersdiscovery.com

is a wonderful website devoted to Johannes Kepler's discoveries.)

__Gérard Desargues__

Desargues invented projective geometry and found the relationship

among conic sections which inspired Blaise Pascal.

Among several ingenious and rigorously proven theorems

are Desargues' Involution Theorem

and his Theorem of Homologous Triangles.

Desargues was also a noted architect and inventor:

he produced an elaborate spiral staircase, invented an

ingenious new pump based on the epicycloid, and had

the idea to use cycloid-shaped teeth in the design of gears.Desargues' projective geometry may have been too creative for his

time; Descartes admired Desargues but was disappointed his friend

didn't apply algebra to his geometric results as Descartes did;

Desargues' writing was poor; and one of his best pupils

(Blaise Pascal himself) turned away from math, so Desargues' work

was largely ignored (except by Philippe de La Hire, Desargues'

other prize pupil) until Poncelet rediscovered it almost two centuries later.

(Copies of Desargues' own works surfaced about the same time.)

For this reason, Desargues may not be important enough

to belong in the Top 100, despite that he may have been among the

greatest natural geometers ever.

__René Descartes__

Descartes' early career was that of soldier-adventurer

and he finished as tutor to royalty, but in between

he achieved fame as the preeminent intellectual of his day.

He is considered the inventor of both analytic geometry and

symbolic algebraic notation and is therefore

called the "Father of Modern Mathematics."

His use of equations to partially solve the geometric Problem of Pappus

revolutionized mathematics.

Because of his famous philosophical writings ("Cogito ergo sum") he

is considered, along with Aristotle, to be one of the

most influential thinkers in history.

He ranks #49 on Michael Hart's famous list of

the Most Influential Persons in History.

His famous mathematical theorems include the

Rule of Signs (for determining the signs of

polynomial roots), the elegant formula

relating the radii ofSoddy kissing circles,

his theorem on total angular defect (an early form of

the Gauss-Bonnet result so key to much mathematics),

and an improved solution to the Delian problem (cube-doubling).

While studying lens refraction,

he invented theOvals of Descartes.

He improved mathematical notation

(e.g. the use of superscripts to denote exponents).

He also discovered Euler's Polyhedral Theorem,F+V = E+2.

Descartes was very influential in physics and biology as well,

e.g. developing laws of motion which included a "vortex" theory of gravitation;

but most of his scientific work outside mathematics was eventually

found to be incorrect.Descartes has an extremely high reputation and would

be ranked even higher by many list makers, but whatever his

historical importance his mathematical skill was not in the

top rank. Some of his work was borrowed from others,

e.g. from Thomas Harriot.

He had only insulting things to say about Pascal and Fermat,

each of whom was much more

brilliant at mathematics than Descartes.

(Some even suspect that Descartes arranged the destruction

of Pascal's lostEssay on Conics.)

And Descartes made numerous errors in his development of

physics, perhaps even delaying science,

with Huygens writing "in all of (Descartes') physics,

I find almost nothing to which I can subscribe as being correct."

Even the historical importance of his mathematics

may be somewhat exaggerated since others, e.g.

Fermat, Wallis and Cavalieri, were making similar discoveries

independently.

__Bonaventura Francesco de Cavalieri__

Cavalieri worked in analysis, geometry

and trigonometry (e.g. discovering a formula for the area of a spherical

triangle), but is most famous for publishing

works on his "principle of indivisibles" (calculus);

these were very influential

and inspired further development by Huygens, Wallis and Barrow.

(His calculus was partly anticipated by Galileo, Kepler and Luca Valerio,

and developed independently, though left unpublished, by Fermat.)

Among his theorems in this calculus was

lim_{(n→∞)}

(1^{m}+2^{m}+ … +n^{m})

/ n^{m+1}= 1 / (m+1)

Cavalieri also worked in theology, astronomy, mechanics and optics;

he was an inventor, and published logarithm tables.

He wrote several books, the first one developing the

properties of mirrors shaped as conic sections.

His name is especially remembered for Cavalieri's

Principle of Solid Geometry.

Galileo said of Cavalieri, "Few, if any, since Archimedes, have delved

as far and as deep into the science of geometry."

__Pierre de Fermat__

Pierre de Fermat was the most brilliant mathematician

of his era and, along with Descartes, one of the most influential.

Although mathematics was just his hobby (Fermat was a government lawyer),

Fermat practically founded Number Theory, and also played key

roles in the discoveries of Analytic Geometry and Calculus.

Lagrange considered Fermat, rather than Newton or Leibniz, to be the

inventor of calculus.

Fermat was first to study certain interesting curves,

e.g. ils"Witch of Agnesi".

He was also an excellent geometer (e.g. discovering a

triangle'sFermat point)

and (in collaboration with Blaise Pascal) discovered probability theory.

Fellow geniuses are the best judges of genius, and Blaise

Pascal had this to say of Fermat:

"For my part, I confess that (Fermat's researches about numbers)

are far beyond me, and I am competent only to admire them."

E.T. Bell wrote "it can be argued that Fermat was

at least Newton's equal as a pure mathematician."Fermat's most famous discoveries in number theory

include the ubiquitously-usedFermat's Little Theorem

(that(ais a multiple of^{p}-a)p

når som helstpis prime);

ilsn = 4case of

his conjecturedFermat's Last Theorem(he may have

proved then = 3case as well);

etFermat's Christmas Theorem

(that any prime (4n+1) can be represented as the sum of two

squares in exactly one way) which may be considered the

most difficult theorem of arithmetic which had been

proved up to that date.

Fermat proved the Christmas Theorem with difficulty

using "infinite descent," but details are unrecorded, so the

theorem is often named theFermat-Euler Prime Number Theorem,

with the first published proof being by Euler more than a century after

Fermat's claim.

Another famous conjecture by Fermat is that

every natural number is the sum of three triangle numbers,

or more generally the sum of k k-gonal numbers.

As with his "Last Theorem" he claimed to have a proof but

didn't write it up.

(This theorem was eventually proved by Lagrange for k=4,

the very young Gauss for k=3, and Cauchy for general k.

Diophantus claimed the k=4 case but any proof has been lost.)

I think Fermat's conjectures were impressive even if unproven,

and that this great mathematician is often underrated.

(Recall that his so-called "Last Theorem"

was actually just a private scribble.)Fermat developed a system of analytic geometry which both

preceded and surpassed that of Descartes; he developed methods of

differential and integral calculus which Newton

acknowledged as an inspiration.

Although Kepler anticipated it, Fermat is credited with Fermat's

Theorem on Stationary Points (df(x)/dx = 0at function extrema),

the key to many problems in applied analysis.

Fermat was also the first European to find the integration

formula for the general polynomial; he used his calculus

to find centers of gravity, etc.Fermat's contemporaneous rival René Descartes is more famous

than Fermat, and Descartes' writings were more influential.

Whatever one thinks of Descartes as aphilosopher,

however, it seems clear that Fermat was the bettermathematician.

Fermat and Descartes did work in physics and independently discovered

the (trigonometric) law of refraction,

but Fermat gave the correct explanation, and used it

remarkably to anticipate the Principle of Least Action

later enunciated by Maupertuis (though Maupertuis

himself, like Descartes, had an incorrect explanation of refraction).

Fermat and Descartes independently discovered analytic geometry,

but it was Fermat who extended it to more than

two dimensions, and followed up by developing elementary calculus.

__Gilles Personne de Roberval__

Roberval was an eccentric genius, underappreciated because

most of his work was published only long after his death.

He did early work in integration, following Archimedes rather

than Cavalieri; he worked on analytic geometry independently of Descartes.

With his analysis he was able to solve several difficult geometric problems

involving curved lines and solids, including results about the

cycloid which were also credited to Pascal and Torricelli.

Some of these methods, published posthumously, led to him being called the

Founder of Kinematic Geometry.

He excelled at mechanics,

worked in cartography, helped Pascal with vacuum experiments, and invented

the Roberval balance, still in use in weighing scales to this day.

He opposed Huygens in the early debate about gravitation,

though neither fully anticipated Newton's solution.

__Evangelista Torricelli__

Torricelli was a disciple of Galileo (and succeeded him

as grand-ducal mathematician of Tuscany).

He was first to understand that a barometer measures atmospheric

weight, and used this insight to invent the mercury barometer and

to create a sustained vacuum (then thought impossible).

(Descartes conjectured, and Pascal later confirmed, that this same

device could also be used as an altimeter.)

Torricelli was a skilled craftsman who built

the best telescopes and microscopes of his day.

As mathematical physicist, he extended Galileo's results, was

first to explain winds correctly, and discovered several key principles

including Torricelli's Law (water drains through a small hole with rate

proportional to the square root of water depth).

In mathematics, he applied Cavalieri's methods to

solve difficult mensuration problems; he also wrote on possible pitfalls

in applying the new calculus.

He discoveredGabriel's Hornwith infinite surface area but finite

volume; this "paradoxical" result provoked much discussion at the time.

(At first Torricelli guessed it was a mistake, just another pitfall

of calculus, though he later accepted its validity.)

He also solved a problem due to Fermat by locating the isogonic center of

a triangle.

Torricelli was a significant influence on the early scientific revolution;

had he lived longer, or published more, he would surely have become

one of the greatest mathematicians of his era.

__John Brehaut Wallis__

Wallis began his life as a savant at arithmetic

(it is said he once calculated the square root of a 53-digit

number to help him sleep and remembered the result in the morning),

a medical student (he may have contributed to the concept

of blood circulation), and theologian, but went on to become perhaps the most

brilliant and influential English mathematician before Newton.

He made major advances in analytic geometry, but also

contributions to algebra, geometry and trigonometry.

Unlike his contemporary Huygens, who took inspiration from

Euclid's rigorous geometry, Wallis embraced the new analytic methods

of Descartes and Fermat.

He is especially famous for using negative and fractional exponents

(though Oresme had introduced fractional exponents three centuries earlier),

taking the areas of curves, and treating inelastic collisions

(he and Huygens were first to develop the law of momentum conservation).

He was a polymath; his non-mathematical work included a highly

respected English grammar; he introduced the (still controversial)

linguistic concept of phonesthesia.He was the first European to solve Pell's Equation.

Like Vieta, Wallis was a code-breaker, helping the Commonwealth

side (though he later petitioned against the beheading of King Charles I).

He was the first great mathematician to consider complex numbers legitimate;

he invented the symbol∞(and used1/∞to denote

infinitesimal).

Wallis coined several terms includingmomentum,

continued fraction,induksjon,interpole,

mantissaethypergeometric series.Also like Vieta, Wallis created an infinite product formula for pi,

which might be (but isn't) written today as:

π = 2 ∏_{k=1,∞}1+(4k^{2}-1)^{-1}

__Blaise Pascal__

Pascal was an outstanding genius who studied geometry as a child.

At the age of sixteen he stated and proved Pascal's Theorem, a

fact relating any six points on any conic section.

The Theorem is sometimes called the "Cat's Cradle"

or the "Mystic Hexagram."

Pascal followed up this result by showing that each of Apollonius'

famous theorems about conic sections was a corollary of the Mystic Hexagram;

along with Gérard Desargues (1591-1661),

he was a key pioneer of projective geometry.

He also made important early contributions to calculus;

indeed it was his writings that inspired Leibniz.

Returning to geometry late in life, Pascal advanced the theory of the cycloid.

In addition to his work in geometry and calculus, he

founded probability theory, and made contributions

to axiomatic theory.

His name is associated with the Pascal's Triangle

of combinatorics and Pascal's Wager in theology.Like most of the greatest mathematicians, Pascal was interested in

physics and mechanics, studying fluids, explaining vacuum,

and inventing the syringe and hydraulic press.

At the age of eighteen he designed and built the world's first automatic

adding machine.

(Although he continued to refine this invention,

it was never a commercial success.)

He suffered poor health throughout his life,

abandoned mathematics for religion at about age 23,

wrote the philosophical treatisePensées

("We arrive at truth, not by reason only, but also by the heart"),

and died at an early age.

Many think that had he devoted more years to mathematics,

Pascal would have been one of the greatest mathematicians ever.

__Christiaan Huygens__

Christiaan Huygens (or Hugens, Huyghens) was second

only to Newton as the greatest mechanist and theoretical

physicist of his era; he may have helped inspire Newton.

Although an excellent mathematician, he

is much more famous for his physical theories and inventions.

He developed laws of motion before Newton, including

the inverse-square law of gravitation, centripetal force,

and treatment of solid bodies rather than point approximations;

he (and Wallis) were first to state the law of momentum conservation correctly.

He advanced the wave ("undulatory") theory of light,

a key concept beingHuygen's Principle, that each point on a wave front

acts as a new source of radiation.

His optical discoveries include explanations for polarization

and phenomena like haloes.

(Because of Newton's high reputation and

corpuscular theory of light, Huygens' superior wave theory

was largely ignored until the 19th-century work of Young, Fresnel,

and Maxwell.

Later, Planck, Einstein and Bohr, partly anticipated by Hamilton,

developed the modern notion of wave-particle duality.)Huygens is famous for his inventions of clocks and lenses.

He invented the escapement and other mechanisms, leading to the

first reliable pendulum clock; he built the first

balance spring watch, which he presented to his

patron, King Louis XIV of France; he was first to give the

correct "equation of time" relating sundial time to

absolute time.

He invented superior lens grinding techniques,

the achromatic eye-piece, and the best telescope of his day.

He was himself a famous astronomer:

he discovered Titan and was first to

properly describe Saturn's rings and the Orion Nebula.

He also designed, but never built, an internal combustion engine.

He promoted the use of an equal-tempered 31-tone music scale

to avoid the tuning errors in Stevin's 12-tone scale;

a 31-tone organ was in use in Holland as late as the 20th century.

Huygens was an excellent card player, billiard player,

horse rider, and wrote a book speculating about extra-terrestrial life.As a mathematician, Huygens did brilliant work in analysis;

his calculus, along with that of Wallis, is considered the

best prior to Newton and Leibniz.

He also did brilliant work in geometry,

proving theorems about conic sections, the cycloid and the catenary.

He was first to show that the cycloid solves the tautochrone

problem; he used this fact to design

pendulum clocks that would be more accurate

than ordinary pendulum clocks.

He was first to find the flaw in Saint-Vincent's

then-famous circle-squaring method;

Huygens himself solved some related quadrature problems.

He introduced the concepts of evolute and involute.

His friendships with Descartes, Pascal, Mersenne and others helped

inspire his mathematics; Huygens in turn was inspirational

to the next generation.

At Pascal's urging, Huygens published the first real

textbook on probability theory; han også

became the first practicing actuary.Huygens had tremendous creativity, historical importance,

and depth and breadth of genius, both in physics and mathematics.

He also was important for serving as tutor

to the otherwise self-taught Gottfried Leibniz

(who'd "wasted his youth" without learning any math).

Before agreeing to tutor him,

Huygens tested the 25-year old Leibniz by asking him

to sum the reciprocals of the triangle numbers.

__Takakazu Seki (Kowa)__

Seki Takakazu (aka Shinsuke) was a self-taught prodigy who

developed a new notation for algebra, and made several discoveries

before Western mathematicians did; disse

include determinants, the Newton-Raphson method,

Newton's interpolation formula,

Bernoulli numbers, discriminants, methods of calculus,

and probably much that has been forgotten

(Japanese schools practiced secrecy).

He calculatedto ten decimal places usingπ

Aitkin's method (rediscovered in the 20th century).

He also worked with magic squares.

He is remembered as a brilliant genius and very influential teacher.Seki's work was not propagated to Europe, so has minimal

historic importance; otherwise Seki might rank high on our list.

__James Gregory__

James Gregory (Gregorie) was the outstanding Scottish genius of his century.

Had he not died at the age of 36, or if he had published more of his work,

(or if Newton had never lived,) Gregory would surely be appreciated

as one of the greatest mathematicians of the early Age of Science.

Inspired by Kepler's work, he worked in mechanics and optics; oppfunnet

a reflecting telescope; and is even

credited with using a bird feather as the first diffraction grating.

But James Gregory is most famous for his mathematics, making

many of the same discoveries as Newton did:

the Fundamental Theorem of Calculus,

interpolation method, and binomial theorem.

He developed the concept of Taylor's series and used it to

solve a famous semicircle division problem posed by Kepler

and to develop trigonometric identities, including

(fortan^{-1}x = x – x^{3}/3 + x^{5}/5 – x^{7}/7 + …|x| )

Gregory anticipated Cauchy's convergence test, Newton's identities

for the powers of roots, and Riemann integration.

He may have been first to suspect that quintics generally lacked

algebraic solutions, as well as thatetπétaite

transcendental.

He produced a partial proof that the ancient "Squaring the Circle"

problem was impossible.

Gregory declined to publish much of his work, partly in deference

to Isaac Newton who was making many of the same discoveries.

Because the wide range of his mathematics wasn't appreciated until long after

his death, Gregory lacks the historic importance to qualify for the Top 100.

__Isaac (Sir) Newton__

Newton was an industrious lad who built marvelous toys

(e.g. a model windmill powered by a mouse on treadmill).

At about age 22, on leave from University, this genius began

revolutionary advances in mathematics, optics, dynamics,

thermodynamics, acoustics and celestial mechanics.

He is famous for his Three Laws of Motion

(inertia, force, reciprocal action) but, as

Newton himself acknowledged, these Laws weren't fully novel:

Hipparchus, Ibn al-Haytham, Descartes, Galileo and Huygens had all

developed much basic mechanics already; and Newton credits the First Law

to Aristotle.

However Newton was apparently the first person to conclude

that the ordinary gravity we observe on Earth is the very same

force that keeps the planets in orbit.

His Law of Universal Gravitation was revolutionary and due to

Newton alone.

(Christiaan Huygens, the other great mechanist of the era,

had independently deduced that Kepler's laws imply inverse-square

gravitation, but he considered the action at a distance in

Newton's theory to be "absurd.")

Newton published the Cooling Law of thermodynamics.

He also made contributions to chemistry, and was

the important early advocate of the atomic theory.

His writings also made important contributions

to the general scientific method.

His other intellectual interests included theology,

and mysticism.

He studied ancient Greek writers like Pythagoras,

Democritus, Lucretius, Plato; and claimed that the

ancients knew much, including the law of gravitation.Although this list is concerned only with mathematics,

Newton's greatness is indicated by the huge range of his

physics: even without his Laws of Motion, Gravitation and Cooling,

he'd be famous just for his revolutionary work in optics, where

he explained diffraction, observed

that white light is a mixture of all the rainbow's colors,

noted that purple is created by combining red and

blue light and, starting from that observation,

was first to conceive of a color hue "wheel."

(The mystery of the rainbow had been solved by earlier

mathematicians like Al-Farisi and Descartes, but

Newton improved on their explanations. Most people

would count only six colors in the rainbow but, due to

Newton's influence, seven — a number with mystic importance —

is the accepted number. Supernumerary rainbows, by the way,

were not explained until the wave theory of light superseded

Newton's theory.)

He noted that his dynamical laws were symmetric in time; que

just as the past determines the future, so the

future might, in principle, determine the past.

Newton almost anticipated Einstein's mass-energy equivalence,

writing "Gross Bodies and Light are convertible into one another…

(Nature) seems delighted with Transmutations."

Ocean tides had intrigued several of Newton's predecessors; en gang

gravitation was known, the Moon's gravitational attraction

provided the explanation — except that there aredeuxhøy

tides per day, one when the Moon is farthest away.

With clear thinking the second high tide is also explained by gravity

but who was the first clear thinker to produce that explanation?

You guessed it! Isaac Newton.

(The theory of tides was later refined by Laplace.)

Newton's earliest fame came when he discovered the problem of

chromatic aberration in lenses, and designed the first reflecting telescope

to counteract that aberration; his were the best telescopes of that era.

He also designed the first reflecting microscope, and the sextant.Although others also developed the techniques independently,

Newton is regarded as the "Father of Calculus" (which he called

"fluxions"); he shares credit with Leibniz for the

Fundamental Theorem of Calculus

(that integration and differentiation are each other's inverse operation).

He applied calculus for several purposes:

finding areas, tangents, the lengths of

curves and the maxima and minima of functions.

Although Descartes is renowned as the inventor of analytic geometry,

he and followers like Wallis were reluctant even to use negative

coordinates, so one historian declares Newton to be "the first

to work boldly with algebraic equations."

In addition to several other important advances in analytic geometry,

his mathematical works include the Binomial Theorem,

his eponymous interpolation method,

the idea of polar coordinates,

and power series for exponential and trigonometric functions.

(The equation

e^{x}=∑x^{k}/ k!

has been attributed to Newton and called the "most important series in

mathematics," but, although he published some related trignometric

formulae, he doesn't seem to have published the exponential

series explicitly priopr to Bernoulli's discoveries circa 1690.)

He contributed to algebra and the theory of equations;

he was first to state Bézout's Theorem;

he generalized Descartes' rule of signs.

(The generalized rule of signs was incomplete and finally

resolved two centuries later by Sturm and Sylvester.)

He developed a series for the arcsin function.

He developed facts about cubic equations

(just as the "shadows of a cone" yield all quadratic curves,

Newton found a curve whose "shadows" yield all cubic curves).

He proved, using a purely geometric argument of awesome ingenuity,

that same-mass spheres (or hollowed spheres) of any radius have equal

gravitational attraction: this fact is key to celestial motions.

(He also proved that objectsà l'intérieura hollowed sphere

experience zero net attraction.)

He discovered Puiseux series almost two centuries before they

were re-invented by Puiseux.

(Like some of the greatest ancient mathematicians,

Newton took the time to compute an approximation to

π; his was better than Vieta's, though still

not as accurate as al-Kashi's.)Newton is so famous for his calculus, optics, and laws of

gravitation and motion, it is easy to overlook that he was also one of the

very greatest geometers.

He was first to fully solve

the famous Problem of Pappus, and did so with pure geometry.

Building on the "neusis" (non-Platonic) constructions of Archimedes

and Pappus, he demonstrated cube-doubling and that angles

kunne værtk-sected for anyk, if one is allowed a conchoid or

certain other mechanical curves.

He also built on Apollonius' famous

theorem about tangent circles to develop the technique

now called hyperbolic trilateration.

Despite the power of Descartes' analytic geometry,

Newton's achievements with synthetic geometry were surpassing.

Even before the invention of the calculus of variations, Newton

was doing difficult work in that field, e.g.

his calculation of the "optimal bullet shape."

His other marvelous geometric theorems included several about quadrilaterals

and their in- or circum-scribing ellipses.

He constructed the parabola defined by four given points,

as well as various cubic curve constructions.

(As with Archimedes, many of

Newton's constructions used non-Platonic tools.)

He anticipated Poncelet's Principle of Continuity.

An anecdote often cited to demonstrate his brilliance is the problem

debrachistochrone, which had baffled the best mathematicians in

Europe, and came to Newton's attention late in life.

He solved it in a few hours and published the answer anonymously.

But on seeing the solution Jacob Bernoulli immediately exclaimed

"I recognize the lion by his footprint."In 1687 Newton published

Philosophiae Naturalis Principia Mathematica, surely

the greatest scientific book ever written.

The motion of the planets was not understood before Newton,

Même siheliocentricsystem allowed Kepler to describe the

orbits.

enPrincipiaNewton analyzed the consequences of his Laws of Motion

and introduced the Law of Universal Gravitation.

With the key mystery of celestial motions finally resolved,

the Great Scientific Revolution began.

(In his work Newton also proved important theorems about inverse-cube

forces, work largely unappreciated until Chandrasekhar's modern-day work.)

Newton once wrote "Truth is ever to be found in the simplicity,

and not in the multiplicity and confusion of things."

Sir Isaac Newton was buried at Westminster Abbey in a tomb

inscribed "Let mortals rejoice that so great an ornament to the human

race has existed."Newton ranks #2 on Michael Hart's famous list of

the Most Influential Persons in History.

(Muhammed the Prophet of Allah is #1.)

Whatever the criteria, Newton would certainly rank first

or second on any list of physicists, or scientists in general,

but some listmakers would demote him slightly on a list of

pure mathematicians:

his emphasis was physics not mathematics,

and the contribution of Leibniz

(Newton's rival for the titleInventor of Calculus)

lessens the historical importance of Newton's calculus.

One reason I've ranked him at #1 is a comment by

Gottfried Leibniz himself:

"Taking mathematics from the beginning of the world to the time when

Newton lived, what he has done is much the better part."

__Gottfried Wilhelm von Leibniz__

Leibniz was one of the most brilliant and prolific

intellectuals ever; and his influence in mathematics (especially

his co-invention of the infinitesimal calculus) was immense.

His childhood IQ has been estimated as second-highest in all of history,

behind only Goethe's.

Descriptions which have been applied to Leibniz include

"one of the two greatest universal geniuses" (da Vinci was

the other); "the most important logician between Aristotle and Boole;"

and the "Father of Applied Science."

Leibniz described himself as "the most teachable of mortals."Mathematics was just a self-taught sideline for Leibniz, who was

a philosopher, lawyer, historian, diplomat and renowned inventor.

Because he "wasted his youth" before learning mathematics,

he probably ranked behind the Bernoullis as well as Newton

in pure mathematical talent, and thus he may be the only

mathematician among the Top Fifteen who was never the greatest

living algorist or theorem prover.

I won't try to summarize Leibniz' contributions to philosophy

and diverse other fields including biology; as just

three examples: he predicted the Earth's molten core,

introduced the notion of subconscious mind,

and built the first calculator that could do multiplication.

Leibniz also had political influence: he

consulted to both the Holy Roman and Russian Emperors;

another of his patrons was Sophia Wittelsbach (Electress of Hanover),

who was only distantly in

line for the British throne, but was made Heir Presumptive.

(Sophia died before Queen Anne, but her son

was crowned King George I of England.)Leibniz pioneered the common discourse of mathematics,

including its continuous, discrete, and symbolic aspects.

(His ideas on symbolic logic weren't pursued and it was left

to Boole to reinvent this almost two centuries later.)

Mathematical innovations attributed to Leibniz include

the notations ∫f(x),réx

,réf(x)/réx∛x,

and even the use of(instead ofa·b

) for multiplication;unX b

the concepts of matrix determinant and Gaussian elimination;

the theory of geometric envelopes;

and the binary number system.

He worked in number theory, conjecturing Wilson's Theorem.

He invented more mathematical terms than anyone, including

funksjon,analysis situ,variable,abscissa,

parameteretkoordinere.

He also coined the wordtranscendental, proving that

sin() was not an algebraic function.

His works seem to anticipate cybernetics and information theory;

and Mandelbrot acknowledged Leibniz' anticipation of self-similarity.

Like Newton, Leibniz discovered The Fundamental Theorem of Calculus;

his contribution to calculus was much more influential than Newton's,

and his superior notation is used to this day.

As Leibniz himself pointed out, since the concept of

mathematical analysis was already known to ancient Greeks,

the revolutionary invention was thenotation("calculus"),

because with "symbols (which) express the exact nature of a

thing briefly … the labor of thought is wonderfully diminished."Leibniz' thoughts on mathematical physics had some influence.

He was one of the first to articulate the law of energy

conservation and may have written on the principle of least action.

He developed laws of motion that gave different insights

from those of Newton; his views on cosmology anticipated theories

of Mach and Einstein and are more in accord with modern

physics than are Newton's views.

Mathematical physicists influenced by Leibniz include not only Mach,

but perhaps Hamilton and Poincaré themselves.Although others found it independently

(including perhaps Madhava three centuries earlier),

Leibniz discovered and proved a striking identity

àπ:

π/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – …

__Jacob Bernoulli__

Jacob Bernoulli studied the works of Wallis and Barrow;

he and Leibniz became friends and tutored each other.

Jacob developed important methods for integral

and differential equations, coining the wordintegral.

He and his brother were the key pioneers in mathematics during the generations

between the era of Newton-Leibniz and the rise of Leonhard Euler.Jacob liked to pose and solve physical optimization problems.

His "catenary" problem (what shape does a clothesline take?)

became more famous than the "tautochrone" solved by Huygens.

Perhaps the most famous of such problems

was the brachistochrone, wherein Jacob recognized

Newton's "lion's paw", and about which Johann Bernoulli wrote:

"You will be petrified with astonishment (that)

this same cycloid, the tautochrone of Huygens,

is the brachistochrone we are seeking."

Jacob did significant work outside calculus;

in fact his most famous work was theArt of Conjecture,

a textbook on probability and combinatorics which

proves the Law of Large Numbers, the Power Series Equation,

and introduces the Bernoulli numbers.

While studying compound interest he introduced the

konstante, though it was given that name by Euler.

He is credited with the invention of polar coordinates (though

Newton and Alberuni had also discovered them).

Jacob also did outstanding work in geometry, for example constructing

perpendicular lines which quadrisect a triangle.

__Johann Bernoulli__

Johann Bernoulli learned from his older brother and Leibniz,

and went on to become principal teacher to Leonhard Euler.

He developed exponential calculus;

together with his brother Jacob, he founded the

calculus of variations.

Johann solved the catenary before Jacob did;

this led to a famous rivalry in the Bernoulli family.

(No joint papers were written; instead the Bernoullis,

especially Johann, began claiming each others' work.)

Although his older brother may have demonstrated greater breadth,

Johann had no less skill than Jacob,

contributed more to calculus,

discovered L'Hôpital's Rule before L'Hôpital did,

and made important contributions in physics, e.g. handle om

vibrations, elastic bodies, optics, tides, and ship sails.It may not be clear which Bernoulli was the "greatest."

Johann has special importance as tutor to Leonhard Euler,

but Jacob has special importance as tutor to his brother Johann.

Johann's son Daniel is also a candidate for greatest Bernoulli.

__Abraham De Moivre__

De Moivre was an important pioneer of analytic geometry

and, especially, probability theory.

(He and Laplace may be regarded as the two most important early

developers of probability theory.)

In probability theory he developed actuarial science, posed interesting

problems (e.g. about derangements), discovered the normal

and Poisson distributions, and proposed (but didn't prove) the

Central Limit Theorem.

He was first to discover a closed-form formula for the Fibonacci numbers;

and he developed an early version of Stirling's approximation ton!.

He discovered De Moivre's Theorem:

(cosx +i sinx)^{n}

=cosnx +i sinnxHe was a close friend and muse of Isaac Newton, who allegedly

told people who asked aboutPrincipia:

"Go to Mr. De Moivre; he knows these things better than I do."

__Brook Taylor__

Brook Taylor invented integration by parts, developed what is now

called the calculus of finite differences, developed

a new method to compute logarithms, made several other key discoveries

of analysis, and did significant work in mathematical physics.

His love of music and painting may have motivated some of his

mathematics:

He studied vibrating strings; et

also wrote an important treatise on perspective in drawing

which helped develop the fields of both projective

and descriptive geometry.

His work in projective geometry rediscovered Desargues' Theorem, introduced

terms likevanishing point, and influenced Lambert.Taylor was one of the few mathematicians of the Bernoulli era

who was equal to them in genius, but his work

was much less influential.

Today he is most remembered for Taylor Series and the associated

Taylor's Theorem, but he shouldn't get full credit for this.

The method had been anticipated by earlier mathematicians including

Gregory, Leibniz, Newton, and, even earlier, Madhava; and was not fully

appreciated until the work of Maclaurin and Lagrange.

__Colin Maclaurin__

Maclaurin received a University degree in divinity

at age 14, with a treatise on gravitation.

He became one of the most brilliant mathematicians

of his era. He wrote extensively on Newton's method of fluxions,

and the theory of equations, advancing these fields;

worked in optics, and other areas of

mathematical physics; but is most noted

for his work in geometry. Lagrange said Maclaurin's geometry was as beautiful

and ingenious as anything by Archimedes. Clairaut, seeing Maclaurin's

methods, decided that he too would prove theorems with geometry

rather than analysis.

Maclaurin did important work on ellipsoids; for his work on tides he shared the

Paris Prize with Euler and Daniel Bernoulli.

As Scotland's top genius, he was called upon for practical work,

including politics.

Although Maclaurin's work was quite influential, his influence

didn't really match his outstanding brilliance:

he failed to adopt Leibnizian calculus with which great progress

was being made on the Continent, and much of his best work

was published posthumously.

Many of his famous results duplicated work by others:

Maclaurin's Series was just a form of Taylor's series; the Euler-Maclaurin

Summation Formula was also discovered by Euler;

and he discovered the Newton-Cotes Integration Formula after Cotes did.

His brilliant results in geometry included the construction

of a conic from five points, but Braikenridge made the same discovery and

published before Maclaurin did.

He discovered the Maclaurin-Cauchy Test for Integral Convergence

before Cauchy did.

He was first to discover Cramer's Paradox, as Cramer himself acknowledged.

Colin Maclaurin found a simpler and more powerful proof of the

fact that the cycloid solves the famous brachistochrone problem.

__Daniel Bernoulli__

Johann Bernoulli had a nephew, three sons and some grandsons who were

all also outstanding mathematicians.

Of these, the most important was his 2nd-oldest son Daniel.

Johann insisted that Daniel study biology and medicine rather than

mathematics, so Daniel specialized initially in

mathematical biology.

He went on to win the Grand Prize of the Paris Academy no less

than ten times, and was a close friend of Euler.

Daniel developed partial differential equations,

preceded Fourier in the use of Fourier series,

did important work in statistics and the theory of equations,

discovered and proved a key theorem about trochoids,

developed a theory of economic risk (motivated by the

St. Petersburg Paradox discovered by his cousin Nicholas),

but is most famous for his key discoveries in mathematical physics:

e.g. the Bernoulli Principle underlying airflight, and

the notion that heat is simply molecules' random kinetic energy.

Daniel Bernoulli is sometimes called the "Founder of Mathematical Physics."

__Leonhard Euler__

Euler may be the most influential mathematician

who ever lived (though some would make him second to Euclid);

he ranks #77 on Michael Hart's famous list of

the Most Influential Persons in History.

His colleagues called him "Analysis Incarnate."

Laplace, famous for denying credit to fellow mathematicians,

once said "Read Euler: he is our master in everything."

His notations and methods in many areas are in use to this day.

Euler was the most prolific mathematician in history

and is often judged to be the best algorist of all time.

(This brief summary can only touch on a few highlights of Euler's work.

The ranking #4 may seem too low for this supreme mathematician,

but Gauss succeeded at proving several theorems which had stumped Euler.)Just as Archimedes extended Euclid's geometry to marvelous heights, so

Euler took marvelous advantage of the analysis

of Newton and Leibniz.

He also gave the world modern trigonometry;

pioneered (along with Lagrange) the calculus of variations;

generalized and proved the Newton-Giraud formulae;

and made important contributions to algebra,

e.g. his study of hypergeometric series.

He was also supreme at discrete mathematics,

inventing graph theory.

He also invented the concept of generating functions; for example,

letting p(n) denote the number of partitions of n, Euler found

the lovely equation:

Σ_{n}p(n) x^{n}

= 1 / Π_{k}(1 – x^{k})

The denominator of the right side here

expands to a series whose exponents all have the(3m^{2}+m)/2

"pentagonal number" form; Euler found an ingenious proof of this.

Euler wrote the first definitive treatise on continued fractions,

establishing several key theorems on that important topic.Euler was a very major figure in number theory: He proved that the

sum of the reciprocals of primes less than x is approx. (ln lnx),

invented the totient function and used it to generalize

Fermat's Little Theorem,

found both the largest then-known prime

and the largest then-known perfect number,

provedeto be irrational,

discovered (though without complete proof) a

broad class of transcendental numbers,

proved that all even perfect numbers

must have the Mersenne number form that Euclid had

discovered 2000 years earlier, and much more.

Euler was also first to prove several interesting theorems

of geometry, including facts about the9-point Feuerbach circle;

relationships among a triangle's altitudes, medians, and

circumscribing and inscribing circles;

the famous Intersecting Chords Theorem;

and an expression for a tetrahedron's volume in terms of its edge lengths.

Euler was first to explore topology, proving theorems

siEuler characteristic, and the famous

Euler's Polyhedral Theorem,F+V = E+2(although it may have

been discovered by Descartes and first proved rigorously by Jordan).

Although noted as the first great "pure mathematician,"

Euler's pump and turbine equations revolutionized the design of pumps;

he also made important contributions to music theory,

acoustics, optics, celestial motions, fluid dynamics, and mechanics.

He extended Newton's Laws of Motion to rotating rigid bodies;

and developed the Euler-Bernoulli beam equation.

On a lighter note, Euler constructed a particularly

"magical" magic square.Euler combined his brilliance with phenomenal concentration.

He developed the first method to estimate the Moon's orbit (the three-body

problem which had stumped Newton), and he settled an arithmetic

dispute involving 50 terms in a long convergent series.

Both these feats were accomplished when he was totally blind.

(About this he said "Now I will have less distraction.")

François Arago said that "Euler calculated without apparent effort,

as men breathe, or as eagles sustain themselves in the wind."Four of the most important constant symbols in mathematics

(,π,e

= √-1, andJe= 0.57721566…)γ

were all introduced or popularized by Euler,

along with operators like.Σ

He did important work with

Riemann's zeta function

ζ(s) = ∑ k^{-s}(although it was not then known

by that name); he anticipated the concept of

analytic continuation by showing

ζ(-1) = 1+2+3+4+… = -1/12.

Euler started as a young student of the Bernoulli family,

and was Daniel Bernoulli's roommate in Saint Petersburg,

where Euler was first employed as a teacher of physiology.

But at age twenty-eight, Euler

discovered the striking identity

ζ(2) =π^{2}/6

This catapulted Euler to instant fame, since the

left-side infinite sum (1 + 1/4 + 1/9 + 1/16 + …)

was a famous problem of the time.

Euler and others developed alternate proofs and generalizations

of this "Basel problem," and of course the ζ (zeta) function

is now very famous.

ici

is an elegant geometric proof for this theorem.

Among many other famous and important identities,

Euler proved the Pentagonal Number Theorem alluded to above

(a beautiful result which

has inspired a variety of discoveries), and the Euler Product Formula

ζ(s) = ∏(1-p^{-s})^{-1}where the right-side product is taken over all primes

p.

His most famous identity (which Richard Feynman

called an "almost astounding … jewel")

unifies the trigonometric and exponential functions:

.e^{i x}=cosx + isinx

(It is almost wondrous how the particular instance

e^{i π}+1 = 0

combines the most important constants and operators together.)Some of Euler's greatest formulae can be combined into curious-looking

formulae forπ:

π^{2}=

6 ζ(2) =

–Logg^{2}(-1) =

6 ∏_{p∈Prime}(1-p^{-2})^{-1/2}

__Alexis Claude Clairaut__

The reputations of Euler and the Bernoullis

are so high that it is easy to overlook that others

in that epoch made essential contributions to mathematical physics.

(Euler made errors in his development of physics, in some cases because

of a Europeanist rejection of Newton's theories in favor of

the contradictory theories of Descartes and Leibniz.)

The Frenchmen Clairaut and d'Alembert were two other great

and influential mathematicians of the mid-18th century.Alexis Clairaut was extremely precocious, delivering a

math paper at age 13, and becoming the youngest person ever

elected to theParis Academy of Sciences.

He developed the concept of skew curves (the earliest precursor

of spatial curvature);

he made very significant contributions in differential

equations and mathematical physics.

Clairaut supported Newton against the Continental schools, and helped

translate Newton's work into French.

The theories of Newton and Descartes gave different predictions

for the shape of the Earth (whether the poles were flattened or

pointy); Clairaut participated in Maupertuis' expedition to Lappland to

measure the polar regions.

Measurements at high latitudes showed the poles to be flattened:

Newton was right.

Clairaut worked on the theories of ellipsoids and the three-body

problem, e.g. Moon's orbit.

That orbit was the major mathematical challenge of the day,

and there was great difficulty reconciling theory and observation.

It was Clairaut who finally resolved this,

by approaching the problem with more rigor than others.

When Euler finally understood Clairaut's solution he called it

"the most important and profound discovery that has ever

been made in mathematics."

Later, when Halley's Comet reappeared as he had predicted,

Clairaut was acclaimed as "the new Thales."

__Jean-Baptiste le Rond d' Alembert__

During the century after Newton, the Laws of Motion needed to be

clarified and augmented with mathematical techniques.

Jean le Rond, named after the Parisian church where he

was abandoned as a baby, played a very key role in that development.

His D'Alembert's Principle clarified Newton's Third

Law and allowed problems in dynamics to be expressed with

simple partial differential equations;

his Method of Characteristics then reduced those equations

àvanligdifferential equations;

to solve the resultant linear systems,

he effectively invented the method of eigenvalues;

he also anticipated the Cauchy-Riemann Equations.

These are the same techniques in use for many problems

in physics to this day.

D'Alembert was also a forerunner in functions of a complex variable,

and the notions of infinitesimals and limits.

With his treatises on dynamics, elastic collisions,

hydrodynamics, cause of winds, vibrating strings,

celestial motions, refraction, etc., the

young Jean le Rond easily surpassed the efforts of his older

rival, Daniel Bernoulli.

He may have been first to speak of time as a "fourth dimension."

(Rivalry with the Swiss mathematicians led to d'Alembert's

sometimes being unfairly ridiculed, although it does seem true that

d'Alembert had very incorrect notions of probability.)D'Alembert was first to prove that every

polynomial has a complex root; this is now called the

Fundamental Theorem of Algebra.

(In France this Theorem is called the D'Alembert-Gauss Theorem.

Although Gauss was first to provide a fully rigorous proof,

d'Alembert's proof preceded, and was more nearly complete

than, the attempted proof by Euler-Lagrange.)

He also did creative

work in geometry (e.g. anticipating Monge's Three Circle Theorem),

and was principal creator of the major encyclopedia of his day.

D'Alembert wrote "The imagination in a mathematician who creates

makes no less difference than in a poet who invents."

__Johann Heinrich Lambert__

(1727-1777) Switzerland, Prussia

Lambert had to drop out of school at age 12

to help support his family, but

went on to become a mathematician of great fame and breadth.

He made key discoveries involving continued fractions that

led him to prove thatπis irrational.

(He proved more strongly

quetan xeteare both^{x}

irrational for any non-zero rationalx.

His proof for this was so remarkable for its time, that its

completeness wasn't recognized for over a century.)

He also conjectured that

etπwere transcendental.e

He made advances in analysis (including the

introduction ofLambert's W function)

and in trigonometry (introducing

the hyperbolic functionssinhetcosh);

proved a key theorem of spherical trigonometry,

and solved the "trinomial equation."

Lambert, whom Kant called "the greatest genius of Germany,"

was an outstanding polymath: In addition to several areas of mathematics,

he made contributions in philosophy, psychology,

cosmology (conceiving of star clusters, galaxies and supergalaxies),

map-making (inventing several distinct map projections),

inventions (he built the first practical hygrometer and photometer),

dynamics, and especially optics (several laws of optics carry his name).Lambert is famous for his work in geometry,

proving Lambert's Theorem (the path of rotation of

a parabola tangent triangle passes through the parabola's focus).

Lagrange declared this famous identity, used to calculate cometary orbits,

to be the most beautiful and significant result in celestial motions.

Lambert was first to explore straight-edge constructions without compass.

He also developed non-Euclidean geometry, long before

Bolyai and Lobachevsky did.

__Joseph-Louis (Comte de) Lagrange__

Joseph-Louis Lagrange (born Giuseppe Lodovico Lagrangia)

was a brilliant man who advanced to

become a teen-age Professor shortly after first studying mathematics.

He excelled in all fields of analysis and number theory;

he made key contributions to the theories of

determinants, continued fractions, and many other fields.

He developed partial differential equations far beyond those

of D. Bernoulli and d'Alembert,

developed the calculus of variations far beyond that of the Bernoullis,

discovered the Laplace transform before Laplace did,

and developed terminology and

notation (e.g. the use off'(x)etf''(x)

for a function's 1st and 2nd derivatives).

He proved a fundamental Theorem of Group Theory.

He laid the foundations for the theory of

polynomial equations which Cauchy, Abel,

Galois and Poincaré would later complete.

Number theory was almost just a diversion for Lagrange,

whose focus was analysis;

nevertheless he was the master of that field as well,

proving difficult and historic theorems including

Wilson's Theorem (pskillelinjer(p-1)! + 1

when p is prime);

Lagrange's Four-Square Theorem (every positive integer is the

sum of four squares);

and thatn·x^{2}+ 1 = y^{2}has solutions for every positive non-square integer

n.Lagrange's many contributions to physics

include understanding of vibrations (he found

an error in Newton's work and published the

definitive treatise on sound), celestial mechanics

(including an explanation of why the Moon keeps the same

face pointed towards the Earth),

ilsPrinciple of Least Action

(which Hamilton compared to poetry), and the discovery of

the Lagrangian points (e.g., in Jupiter's orbit).

Lagrange's textbooks were noted for clarity and inspired

most of the 19th-century mathematicians on this list.

Unlike Newton, who used calculus to derive his results but then worked

backwards to create geometric proofs for publication, Lagrange

relied only on analysis.

"No diagrams will be found in this work" he wrote in the preface to his

mesterverkMécanique analytique.Lagrange once wrote "As long as algebra and geometry have been

separated, their progress have been slow and their uses limited;

but when these two sciences have been united, they have

lent each mutual forces, and have marched together towards perfection."

Both W.W.R. Ball and E.T. Bell, renowned mathematical historians,

bypass Euler to name Lagrange as "the Greatest

Mathematician of the 18th Century."

Jacobi bypassed Newton and Gauss to call Lagrange "perhaps the greatest

mathematical genius since Archimedes."

__Gaspard Monge (Comte de Péluse)__

Gaspard Monge, son of a humble peddler,

was an industrious and creative inventor

who astounded early with his genius, becoming a professor of

physics at age 16.

As a military engineer he developed the new field of descriptive geometry,

so useful to engineering that it was kept a military secret for 15 years.

Monge made early discoveries in chemistry and helped promote

Lavoisier's work;

he also wrote papers on optics and metallurgy;

Monge's talents were so diverse that he became Minister of the Navy

in the revolutionary government, and eventually became a close friend

and companion of Napoleon Bonaparte.

Traveling with Napoleon he demonstrated great courage

on several occasions.In mathematics, Monge is called the "Father of Differential Geometry,"

and it is that foundational work for which he is most praised.

He also did work in discrete math, partial differential equations,

and calculus of variations.

He anticipated Poncelet's Principle of Continuity.

Monge's most famous theorems of geometry are the Three Circles Theorem

and Four Spheres Theorem.

His early work in descriptive geometry has little interest to pure

mathematics, but his application of calculus to the

curvature of surfaces inspired Gauss and eventually Riemann, and led

the great Lagrange to say "With (Monge's) application

of analysis to geometry this devil of a man will make

himself immortal."Monge was an inspirational teacher whose students included

Fourier, Chasles, Brianchon, Ampere, Carnot,

Poncelet, several other famous mathematicians, and perhaps

indirectly, Sophie Germain.

Chasles reports that Monge never drew figures in his lectures,

but could make "the most complicated forms appear in space …

with no other aid than his hands, whose movements admirably supplemented

his words."

The contributions of Poncelet to synthetic geometry

may be more important than those

of Monge, but Monge demonstrated great genius as an untutored child,

while Poncelet's skills probably developed due to his great teacher.

__Pierre-Simon (Marquis de) Laplace__

Laplace was the preeminent mathematical astronomer,

and is often called the "French Newton."

His masterpiece wasMécanique Céleste

which redeveloped and improved Newton's work on planetary motions

using calculus.

While Newton had shown that the two-body gravitation problem

led to orbits which were ellipses (or other conic sections),

Laplace was more interested in the much more difficult problems

involving three or more bodies. (Would Jupiter's pull on Saturn

eventually propel Saturn into a closer orbit, or was

Saturn's orbit stable for eternity?)

Laplace's equations had the optimistic outcome that the solar

system was stable.Laplace advanced the nebular hypothesis of solar system origin, and

was first to conceive of black holes.

(He also conceived of multiple galaxies,

but this was Lambert's idea first.)

He explained the so-called secular acceleration of the Moon.

(Today we know Laplace's theories do not

fully explain the Moon's path, nor guarantee orbit stability.)

His other accomplishments in physics include theories about

the speed of sound and surface tension.

He worked closely with Lavoisier, helping to discover the elemental

composition of water, and the natures

of combustion, respiration and heat itself.

Laplace may have been first to note that the laws of mechanics

are the same with time's arrow reversed.

He was noted for his strong belief in determinism, famously replying to

Napoleon's question about God with: "I have no need of that hypothesis."Laplace viewed mathematics as just a tool for developing his

physical theories.

Nevertheless, he made many important mathematical discoveries

and inventions (although the Laplace Transform itself

was already known to Lagrange).

He was the premier expert at differential and difference equations,

and definite integrals.

He developed spherical harmonics, potential theory, and the

theory of determinants; anticipated Fourier's series;

and advanced Euler's technique of generating functions.

In the fields of probability and statistics he made key advances:

he proved the Law of Least Squares,

and introduced the controversial ("Bayesian") rule of succession.

In the theory of equations, he was first to prove that

any polynomial of even degree must have a real quadratic factor.Others might place Laplace higher on the List,

but he proved no fundamental theorems ofrenmathematics

(though his partial differential equation for fluid dynamics

is one of the most famous in physics),

founded no major branch of pure mathematics,

and wasn't particularly concerned with rigorous proof.

(He is famous for skipping difficult proof steps with the

phrase "It is easy to see".)

Nevertheless he was surely one of the

størstanvendtmathematicians ever.

__Adrien Marie Legendre__

Legendre was an outstanding mathematician who

did important work in plane and solid geometry,

spherical trigonometry,

celestial mechanics and other areas of physics,

and especially elliptic integrals and number theory.

He found key results in the theories of sums of squares and

sums of k-gonal numbers.

(For example, he showed that all integers except4^{k}(8m+7)

can be expressed as the sum of three squares.)

He also made key contributions in several areas of analysis:

he invented the Legendre transform and Legendre polynomials;

the notation for partial derivatives is due to him.

He invented the Legendre symbol; invented the study of zonal harmonics;

proved thatπétait^{2}

irrational (the irrationality ofπ

had already been proved by Lambert);

and wrote important textbooks in several fields.

Although he never accepted non-Euclidean geometry,

and had spent much time trying to prove the Parallel Postulate,

his inspiring geometry text remained a standard until the 20th century.

As one of France's premier mathematicians, Legendre did other

significant work, promoting the careers of Lagrange and Laplace,

developing trig tables, geodesic projects, etc.There are several important theorems proposed by Legendre

for which he is denied credit, either because his proof was

incomplete or was preceded by another's.

He proposed the famous theorem about primes in a

progression which was proved by Dirichlet; proved and used the

Law of Least Squares which Gauss had left unpublished;

proved the N=5 case of Fermat's Last Theorem which is credited

to Dirichlet; proposed the famous Prime Number Theorem

which was finally proved by Hadamard;

improved the Fermat-Cauchy result about

sums of k-gonal numbers but this topic wasn't fruitful;

and developed various techniques commonly credited to Laplace.

His two most famous theorems of number theory,

the Law of Quadratic Reciprocity

and the Three Squares Theorem (a difficult extension of Lagrange's

Four Squares Theorem), were each enhanced by Gauss a few years

after Legendre's work.

Legendre also proved an early version of Bonnet's Theorem.

Legendre's work in the theory of equations and

elliptic integrals directly inspired the achievements of Galois and Abel

(which then obsoleted much of Legendre's own work);

Chebyshev's work also built on Legendre's foundations.

__Jean Baptiste Joseph
Fourier__

Joseph Fourier had a varied career:

precocious but mischievous orphan,

theology student, young professor of mathematics

(advancing the theory of equations), then revolutionary activist.

Under Napoleon he was a brilliant and important teacher

and historian; accompanied the French Emperor to Egypt;

and did excellent service as district governor of Grenoble.

In his spare time at Grenoble he continued the work

in mathematics and physics that led to his immortality.

After the fall of Napoleon, Fourier exiled himself to England,

but returned to France when offered an important academic

position and published his revolutionary treatise on the

Theory of Heat.

Fourier anticipated linear programming, developing

the simplex method and Fourier-Motzkin Elimination;

and did significant work in operator theory.

He is also noted for the notion of dimensional analysis,

was first to describe the Greenhouse Effect, and continued

his earlier brilliant work with equations.Fourier's greatest fame rests on

his use of trigonometric series (now calledFourier series)

in the solution of differential equations.

Since "Fourier" analysis is in extremely common use among

applied mathematicians, he joins the select company of the

eponyms of "Cartesian" coordinates, "Gaussian" curve, and

"Boolean" algebra.

Because of the importance of Fourier analysis,

many listmakers would rank Fourier much higher than I have done;

however the work was not exceptional asrenmathematics.

Fourier's Heat Equation built on Newton's Law of Cooling;

and the Fourier series solution itself

had already been introduced by Euler, Lagrange and Daniel Bernoulli.Fourier's solution to the heat equation was counterintuitive

(heat transfer doesn't seem to involve the oscillations fundamental

to trigonometric functions): The brilliance of Fourier's imagination

is indicated in that the solution had beenavvist

by Lagrange himself.

Although rigorous Fourier Theorems were finally proved

only by Dirichlet, Riemann and Lebesgue,

it has been said that it was Fourier's

"very disregard for rigor" that led to his great achievement,

which Lord Kelvin compared to poetry.

__Johann Carl Friedrich Gauss__

Carl Friedrich Gauss, the "Prince of Mathematics," exhibited

his calculative powers when he corrected his father's arithmetic before

the age of three. His revolutionary nature was demonstrated at age twelve,

when he began questioning the axioms of Euclid. His genius was confirmed

at the age of nineteen when he proved that the regular n-gon was constructible

if and only if it is the product of distinct prime Fermat numbers.

(He didn't complete the proof of the only-if part.

Cliquez

to see construction of regular 17-gon.)

Also at age 19, he proved Fermat's conjecture that every number is

the sum of three triangle numbers.

(He further determined the number of distinct ways such a sum could be formed.)

At age 24 he publishedDisquisitiones Arithmeticae, probably the

greatest book of pure mathematics ever.Although he published fewer papers than some other

great mathematicians, Gauss may be the greatest theorem prover ever.

Several important theorems and lemmas bear his name;

his proof of Euclid's Fundamental Theorem of Arithmetic

(Unique Prime Factorization) is considered the first rigorous proof;

he extended this Theorem to the Gaussian (complex) integers;

and he was first to produce a rigorous

proof of the Fundamental Theorem of Algebra

(that an n-th degree polynomial has n complex roots);

his Theorema Egregium ("Remarkable Theorem") that

a surface's essential curvature

derived from its 2-D geometry laid the foundation of differential geometry.

Gauss himself used "Fundamental Theorem" to refer to

Euler's Law of Quadratic Reciprocity; Gauss was first to provide

a proof for this, and provided eight distinct proofs for it

over the years.

Gauss proved the n=3 case of Fermat's Last Theorem for Eisenstein

integers (the triangular lattice-points on the complex plane);

though more general, Gauss' proof was simpler than the real integer proof;

this simplification method revolutionized algebra.

He also found a simpler proof for Fermat's Christmas Theorem, by taking

advantage of the identityx.^{2}+y^{2}= (x + iy)(x – iy)

Other work by Gauss led to fundamental theorems in statistics, vector

analysis, function theory, and generalizations of

the Fundamental Theorem of Calculus.Gauss built the theory of complex numbers into its modern form, including

the notion of "monogenic" functions

which are now ubiquitous in mathematical physics.

(Constructing the regular 17-gon as a teenager was actually an

exercise in complex-number algebra, not geometry.)

Gauss developed the arithmetic of congruences and became

the premier number theoretician of all time.

Other contributions of Gauss include

hypergeometric series,

foundations of statistics, and differential geometry.

He also did important work in geometry,

providing an improved solution to Apollonius' famous problem

of tangent circles, stating and

proving theFundamental Theorem of Normal Axonometry,

and solving astronomical problems related to comet

orbits and navigation by the stars.

Ceres, the first asteroid, was discovered when Gauss was a young man;

but only a few observations were made before it disappeared into the

Sun's brightness. Could its orbit be predicted well enough

to rediscover it on re-emergence? Laplace, one of the most

respected mathematicians of the time, declared it impossible.

Gauss became famous when he used an 8th-degree polynomial equation

to successfully predict Ceres' orbit.

Gauss also did important work in several areas of physics,

developed an important modification to Mercator's map projection,

invented the heliotrope, and co-invented the telegraph.Much of Gauss's work wasn't published: unbeknownst to his colleagues it was

Gauss who first discovered non-Euclidean geometry (even

anticipating Einstein by suggesting physical space

might not be Euclidean),

doubly periodic elliptic functions,

a prime distribution formula,

quaternions, foundations of topology, the Law of Least Squares,

Dirichlet's class number formula,

the key Bonnet's Theorem of differential geometry

(now usually called Gauss-Bonnet Theorem),

the butterfly procedure for rapid calculation of Fourier series,

and even the rudiments of knot theory.

Gauss was first to prove the Fundamental Theorem of Functions of

a Complex Variable (that the line-integral over a closed

curve of a monogenic function is zero),

but he let Cauchy take the credit.

Gauss was very prolific, and may be the most

brilliant theorem prover who ever lived, so many would rank him #1.

But several others on the list had morehistoricalimportance.

Abel hints at a reason for this:

"(Gauss) is like the fox, who effaces his tracks in the sand."Gauss once wrote "It is not knowledge, but the act of learning, …

which grants the greatest enjoyment.

When I have clarified and exhausted a subject,

then I turn away from it, in order to go into darkness again …"

__Siméon Denis Poisson__

Siméon Poisson was a protégé of Laplace and,

like his mentor, is

among the greatest applied mathematicians ever.

Poisson was an extremely prolific researcher and also an excellent

teacher.

In addition to important advances in several areas of physics,

Poisson made key contributions to Fourier analysis, definite

integrals, path integrals, statistics, partial differential equations,

calculus of variations and other fields of mathematics.

Dozens of discoveries are named after Poisson; par exemple

the Poisson summation formula which has applications in analysis,

number theory, lattice theory, etc.

He was first to note the paradoxical properties of the Cauchy distribution.

He made improvements to Lagrange's equations of celestial

motions, which Lagrange himself found inspirational.

Another of Poisson's contributions to mathematical

physics was his conclusion that the wave theory of light

implies a brightArago spotat the center of certain shadows.

(Poisson used this paradoxical result to argue that

the wave theory was false, but instead the Arago spot,

hitherto hardly noticed, was observed experimentally.)

Poisson once said "Life is good for only two things,

discovering mathematics and teaching mathematics."

__Bernard Placidus Johann Nepomuk Bolzano__

Bolzano was an ordained Catholic priest, a religious philosopher,

and focused his mathematical attention on fields like metalogic,

writing "I prized only … mathematics which was … philosophy."

Still he made several important mathematical discoveries ahead of his time.

His liberal religious philosophy upset the Imperial rulers; han

was charged with heresy, placed under house arrest, and his

writings censored. This censorship meant that

many of his great discoveries turned up

only posthumously, and were first rediscovered by others.

He was noted for advocating great rigor, and is appreciated

for developing theapproach for(ε, δ)

rigorous proofs in analysis; this work inspired the great Weierstrass.Bolzano gave the first analytic proof of the Fundamental Theorem

of Algebra; the first rigorous proof that continuous functions

achieve any intermediate value (Bolzano's Theorem, rediscovered

by Cauchy);

the first proof that a bounded sequence of reals has a convergent

subsequence (Bolzano-Weierstrass theorem);

was first to describe a nowhere-differentiable continuous function;

and anticipated Cantor's discovery of the distinction

between denumerable and non-denumerable infinities.

If he had focused on mathematics and published more, he might be

considered one of the most important mathematicians of his era.

__Jean-Victor Poncelet__

After studying under Monge, Poncelet became an

officer in Napoleon's army, then a prisoner of the Russians.

To keep up his spirits as a prisoner he devised and solved

mathematical problems using charcoal and the walls of his

prison cell instead of pencil and paper.

During this time he reinvented projective geometry.

Regaining his freedom, he wrote many papers, made numerous

contributions to geometry;

he also made contributions to practical mechanics.

Poncelet is considered one of the most influential

geometers ever; he is especially noted for his

Principle of Continuity, an intuition with broad application.

His notion of imaginary solutions in geometry was inspirational.

Although projective geometry had been studied earlier

by mathematicians like Desargues, Poncelet's work

excelled and served as an inspiration for other

branches of mathematics including algebra,

topology, Cayley's invariant

theory and group-theoretic developments by Lie and Klein.

His theorems of geometry include his Closure Theorem

about Poncelet Traverses, the Poncelet-Brianchon

Hyperbola Theorem, and Poncelet's Porism (if two conic

sections are respectively inscribed and circumscribed by

an n-gon, then there are infinitely many such n-gons).

Perhaps his most famous theorem, although it was left to

Steiner to complete a proof, is the beautiful Poncelet-Steiner

Theorem about straight-edge constructions.

__Augustin-Louis Cauchy__

Cauchy was extraordinarily prodigious, prolific and inventive.

Home-schooled, he awed famous mathematicians at an early age.

In contrast to Gauss and Newton, he was almost over-eager to publish;

in his day his fame surpassed that of Gauss and has continued to grow.

Cauchy did significant work in analysis, algebra, number theory

and discrete topology.

His most important contributions included convergence criteria for

infinite series, the "theory of substitutions"

(permutation group theory), and especially

his insistence on rigorous proofs.Cauchy's research also included

differential equations, determinants, and probability.

He invented the calculus of residues, rediscovered Bolzano's Theorem,

and much more.

Although he was one of the first great mathematicians

to focus on abstract mathematics (another was Euler),

he also made important contributions to mathematical physics, e.g. ils

theory of elasticity.

Cauchy's theorem of solid geometry is important in rigidity theory;

the Cauchy-Schwarz Inequality has very wide application

(e.g. as the basis for Heisenberg's Uncertainty Principle);

several important lemmas of analysis are due to Cauchy;

the famous Burnside's Counting Theorem was first discovered by Cauchy; etc.

He was first to prove Taylor's Theorem rigorously, and

first to prove Fermat's conjecture that every

positive integer can be expressed

as the sum of k k-gonal numbers for any k.

(Gauss had proved the case k = 3.)One of the duties of a great mathematician is to nurture

his successors, but Cauchy selfishly dropped the ball

on both of the two greatest young mathematicians of his day,

mislaying key manuscripts ofles deuxAbel and Galois.

Cauchy is credited with group theory, yet it was Galois who

invented this first, abstracting it far more than Cauchy did,

some of this in a work which Cauchy "mislaid."

(For this historicalmiscontributionperhaps Cauchy

should be demoted.)

__August Ferdinand Möbius__

Möbius worked as a Professor of physics and astronomy,

but his astronomy teachers included Carl Gauss and other brilliant

mathematicians, and Möbius is most noted for his work in mathematics.

He had outstanding intuition and originality, and prepared his

books and papers with great care.

He made important advances in number theory, topology,

and especially projective geometry.

Several inventions are named after him, such as the Möbius

transformation and Möbius net of geometry, and

the Möbius function and Möbius inversion formula

of algebraic number theory.

He is most famous for the Möbius strip; this one-sided

strip was first discovered by Lister, but Möbius went much

further and developed important new insights in topology.Möbius' greatest contributions were to projective geometry,

where he introduced the use of

homogeneous barycentric coordinates as well as signed angles and lengths.

These revolutionary discoveries inspired Plücker, and were

declared by Gauss to be

"among the most revolutionary intuitions in the history of mathematics."

__Nicolai Ivanovitch Lobachevsky__

Lobachevsky is famous for discovering non-Euclidean geometry.

He did not regard this new geometry as simply a theoretical

curiosity, writing "There is no branch of mathematics … which may

not someday be applied to the phenomena of the real world."

He also worked in several branches of analysis and physics, anticipated the

modern definition of function, and may have been first to explicitly note the

distinction between continuous and differentiable curves.

He also discovered the important Dandelin-Gräffe method of

polynomial roots independently of Dandelin and Gräffe.

(In his lifetime, Lobachevsky was under-appreciated and over-worked;

his duties led him to learn architecture and even some medicine.)Although Gauss and Bolyai discovered non-Euclidean geometry

independently about the same time as Lobachevsky, it is worth noting

that both of them had strong praise for Lobachevsky's genius.

His particular significance was in daring to reject a 2100-year old axiom;

thus William K. Clifford called Lobachevsky "the Copernicus of Geometry."

__Michel Floréal Chasles__

Chasles was a very original thinker who developed new

techniques for synthetic geometry. He introduced new notions

quiblyantetcross-ratio;

made great progress with thePrinciple of Duality;

and showed how to combine the power of analysis with the intuitions

of geometry.

He invented atheory of characteristicsand used it

to become the Founder of Enumerative Geometry.

He proved a key theorem about solid body kinematics.

His influence was very large; for example Poincaré

(student of Darboux, who in turn was Chasles' student)

often applied Chasles' methods.

Chasles was also a historian of mathematics; for example he noted

that Euclid had anticipated the method of cross-ratios.

__Jakob Steiner__

Jakob Steiner made many major advances in synthetic geometry, hoping that

classical methods could avoid any need for analysis;

and indeed, like Isaac Newton, he was often able to equal or surpass methods

of analysis or the calculus of variations using just pure geometry;

for example he had pure synthetic proofs for a notable extension to Pascal's Mystic Hexagram,

and a reproof of Salmon's Theorem that cubic surfaces have exactly 27 lines.

(He wrote "Calculating replaces thinking while geometry stimulates it.")

One mathematical historian (Boyer) wrote "Steiner reminds one of Gauss

in that ideas and discoveries thronged

through his mind so rapidly that he could scarcely reduce them to order on paper."

Although thePrinciple of Dualityunderliggende

projective geometry was already known, he gave it a radically

new and more productive basis, and created a new theory of conics.

His work combined generality, creativity and rigor.Steiner developed several famous construction methods, e.g.

for a triangle's smallest circumscribing and largest inscribing ellipses,

and for its "Malfatti circles."

Among many famous and important

theorems of classic and projective geometry,

he proved that the

Wallace lines of a triangle lie in a 3-pointed hypocycloid,

developed a formula for the partitioning of space by planes,

a fact about the surface areas of tetrahedra,

and proved several facts about his famous

Steiner's Chain of tangential circles and his famous "Roman surface."

Perhaps his three most famous theorems are

the Poncelet-Steiner Theorem (lengths constructible

with straightedge and compass can be constructed with straightedge

alone as long as the picture plane contains the center

and circumference of some circle), the Double-Element Theorem

about self-homologous elements in projective geometry,

and the Isoperimetric Theorem that among solids of equal

volume the sphere will have minimum area, etc.

(Dirichlet found a flaw in the proof of the Isoperimetric Theorem

which was later corrected by Weierstrass.)

Steiner is often called,

along with Apollonius of Perga (who lived 2000 years earlier),

one of the two greatest pure geometers ever.

(The qualifier "pure" is added to exclude such geniuses

as Archimedes, Newton and Pascal from this comparison.

I've included Steiner for his extreme brilliance and productivity:

several geometers had much more historic influence, and assolely

a geometer he arguably lacked "depth.")Steiner once wrote:

"For all their wealth of content, … music, mathematics, and chess

are resplendently useless (applied mathematics is a higher plumbing,

a kind of music for the police band). They are metaphysically trivial,

irresponsible. They refuse to relate outward, to take reality for arbiter.

This is the source of their witchery."

__Julius Plücker__

Plücker was one of the most innovative geometers,

inventing line geometry (extending the atoms of geometry

beyond just points), enumerative geometry (which considered

such questions as the number of loops in an algebraic curve),

geometries of more than three dimensions, and generalizations

of projective geometry.

He also gave an improved theoretic basis for the Principle of Duality.

His novel methods and notations were important to the development

of modern analytic geometry, and inspired Cayley, Klein and Lie.

He resolved the famous Cramer-Euler Paradox and the related

Poncelet Paradox by studying the singularities of curves;

Cayley described this work

as "most important … beyond all comparison in the entire subject

of modern geometry."

In part due to conflict with his more famous rival, Jakob Steiner,

Plücker was under-appreciated in his native Germany,

but achieved fame in France and England.

In addition to his mathematical work in algebraic and

analytic geometry,

Plücker did significant work in physics, e.g.

his work with cathode rays.

Although less brilliant as a theorem prover than

Steiner, Plücker's work, taking

full advantage of analysis and seeking physical applications,

was far more influential.

__Niels Henrik Abel__

At an early age, Niels Abel studied the works of the

greatest mathematicians, found flaws in their proofs, and resolved to

reprove some of these theorems rigorously.

He was the first to fully prove the general case of Newton's

Binomial Theorem, one of the most widely applied theorems in mathematics.

Several important theorems of analysis are named after Abel,

including the (deceptively simple) Abel's Theorem of

Convergence (published posthumously).

Along with Galois, Abel is considered one of the two founders of group theory.

Abel also made contributions in algebraic geometry

and the theory of equations.

Inversion

(replacingy = f(x)avecx = f)^{-1}(y)

is a key idea in mathematics (consider Newton's Fundamental Theorem

of Calculus); Abel developed this insight.

Legendre had spent much of his life studying elliptic integrals,

but Abel inverted these to get elliptic functions,

and was first to observe (but in a manuscript mislaid by

Cauchy) that they were doubly periodic.

Elliptic functions quickly became a productive field of mathematics,

and led to more general complex-variable functions,

which were important to the development of

both abstract and applied mathematics.Finding the roots of polynomials is a key mathematical

problem: the general solution of the quadratic equation was

known by ancients; the discovery of general methods for

solving polynomials of degree three and four is usually treated

as the major math achievement of the 16th century; si

for over two centuries

an algebraic solution for the general 5th-degree polynomial

(quintic) was a Holy Grail

sought by most of the greatest mathematicians.

Abel proved that most quintics didpashave such solutions.

This discovery, at the age of only nineteen, would have quickly

awed the world, but Abel was impoverished, had few contacts,

and spoke no German.

When Gauss received Abel's manuscript he discarded it

unread, assuming the unfamiliar author was just another crackpot trying to

square the circle or some such.

His genius was too great for him to be ignored long,

but, still impoverished,

Abel died of tuberculosis at the age of twenty-six.

His fame lives on and even the lower-case word

'abelian' is applied to several concepts.

Liouville said Abel was the greatest genius he ever met.

Hermite said "Abel has left mathematicians enough to keep them busy

for 500 years."

__Carl G. J. Jacobi__

Jacobi was a prolific mathematician who

did decisive work in the algebra and analysis of complex variables,

and did work in number theory

(e.g. cubic reciprocity) which excited Carl Gauss.

He is sometimes described as the successor to Gauss.

As an algorist (manipulator of involved algebraic expressions),

he may have been surpassed only by Euler and Ramanujan.

He was also a very highly regarded teacher.

In mathematical physics, Jacobi perfected Hamilton's

principle of stationary action, and made other important advances.Jacobi's most significant early achievement was the theory

of elliptic functions, e.g. his fundamental result about

functions with multiple periods.

Jacobi was the first to apply elliptic functions to number theory,

extending Lagrange's famous Four-Squares Theorem to show

enhow many distinct ways

a given integer can be expressed as the sum of four squares.

He also made important discoveries in many other areas including

theta functions (e.g. his Jacobi Triple Product Identity),

higher fields, number theory, algebraic geometry,

differential equations, q-series, hypergeometric series,

determinants, Abelian functions, and dynamics.

He devised the algorithms still used to calculate eigenvectors

and for other important matrix manipulations.

The range of his work is suggested by the fact that the

"Hungarian method," an efficient solution to an optimization

problem published more than a century after Jacobi's death,

has since been found among Jacobi's papers.Like Abel, as a young man

Jacobi attempted to factor the general quintic equation.

Unlike Abel, he seems never to have considered proving its

impossibility.

This fact is sometimes cited to show that despite Jacobi's

creativity, his ill-fated contemporary was the more brilliant genius.

__Johann Peter Gustav
Lejeune Dirichlet__

Dirichlet was preeminent in algebraic and analytic

number theory, but did advanced work in several other fields as well:

He discovered the modern definition of function,

the Voronoi diagram of geometry, and important concepts in

differential equations, topology, and statistics.

His proofs were noted both for great ingenuity and unprecedented rigor.

As an example of his careful rigor, he found a fundamental flaw

in Steiner's Isoperimetric Theorem proof which no one else had noticed.

In addition to his own discoveries, Dirichlet played a key role in

interpreting the work of Gauss,

and was an influential teacher,

mentoring famous mathematicians like

Bernhard Riemann (who considered Dirichlet second only to Gauss

among living mathematicians),

Leopold Kronecker and Gotthold Eisenstein.As an impoverished lad Dirichlet spent his money on

math textbooks; Gauss' masterwork became his life-long companion.

Fermat and Euler had proved the impossibility of

x^{k}+ y^{k}= z^{k}for k = 4 and k = 3; Dirichlet became

famous by proving impossibility for k = 5 at the age of 20.

Later he proved the case k = 14 and, later still, may have helped

Kummer extend Dirichlet's quadratic fields, leading to proofs of

more cases.

More important than his work with Fermat's Last Theorem

was his Unit Theorem, considered one of the most important

theorems of algebraic number theory.

The Unit Theorem is unusually difficult to prove;

it is said that Dirichlet discovered the proof while listening to music

in the Sistine Chapel.

A key step in the proof usesDirichlet's Pigeonhole Principle,

a trivial idea but which Dirichlet applied with great ingenuity.Dirichlet did seminal work in analysis and is

considered the founder of analytic number theory.

He invented a method of L-series to prove the important

theorem (Gauss' conjecture)

that any arithmetic series (without a common factor) has an

infinity of primes.

It was Dirichlet who proved the fundamental Theorem of Fourier

series: that periodic analytic functions

can always be represented as a simple

trigonometric series.

Although he never proved it rigorously, he is especially noted

pourDirichlet's Principlewhich posits the existence

of certain solutions in the calculus of variations,

and which Riemann found to be particularly fruitful.

Other fundamental results Dirichlet contributed to analysis and

number theory include

a theorem about Diophantine approximations

and hisClass Number Formula.

__William Rowan (Sir) Hamilton__

Hamilton was a childhood prodigy.

Home-schooled and self-taught,

he started as a student of languages and literature,

was influenced by an arithmetic prodigy his own age,

read Euclid, Newton and Lagrange, found an error

by Laplace, and made new discoveries in optics;

all this before the age of seventeen when he first

attended school.

At college he enjoyed unprecedented success

in all fields, but his

undergraduate days were cut short abruptly by his

appointment as Royal Astronomer of Ireland at the age of 22.

He soon began publishing his revolutionary treatises on optics,

in which he developed Hamilton's Principle of Stationary Action.

This Principle refined and corrected the earlier principles of

least action developed by Maupertuis, Fermat, and Euler;

it (and related principles) are key to much of modern physics.

His early writing also predicted that some crystals would have an hitherto

unknown "conical" refraction mode; this was soon confirmed experimentally.Hamilton's Principle of Least Action, and its associated equations

and concept of configuration space, led to

a revolution in mathematical physics.

Since Maupertuis had named this Principle a century

earlier, it is possible to underestimate Hamilton's contribution.

However Maupertuis, along with others credited with anticipating

the idea (Fermat, Leibniz, Euler and Lagrange) failed to

state the full Principle correctly.

Rather than minimizing action, physical systems sometimes achieve

a non-minimal butstasjonæraction in configuration space.

(Poisson and d' Alembert had noticed exceptions to Euler-Lagrange

least action, but failed to find Hamilton's solution.

Jacobi also deserves some credit for the Principle, but his

work came after reading Hamilton.)

Because of this Principle, as well as his wave-particle duality

(which would be further developed by Planck

and Einstein), Hamilton

can be considered a major early influence on quantum theory.Hamilton also made revolutionary contributions

to dynamics, differential equations, the theory of equations,

numerical analysis, fluctuating functions,

and graph theory (he marketed a puzzle based on hisHamiltonian paths).

He invented the ingenious hodograph.

He coined several mathematical terms including

vector,scalar,associativeettensor.

In addition to his brilliance and creativity, Hamilton was

renowned for thoroughness and produced voluminous writings

on several subjects.Hamilton himself considered his greatest accomplishment

to be the development of quaternions, a non-Abelian field to

handle 3-D rotations.

While there is no 3-D analog to the

Gaussian complex-number plane

(based on the equation)Je^{2}= -1

quaternions derive from a 4-D analog based on

.Je^{2}=j^{2}=

k^{2}=ijk= –jik= -1

Although matrix and tensor methods may seem more general,

quaternions are still in wide engineering use because of

practical advantages, e.g. avoidance of "gimbal lock."Hamilton once wrote:

"On earth there is nothing great but man;

in man there is nothing great but mind."

__Hermann Günter Grassmann__

Grassmann was an exceptional polymath:

begrepetGrassmann's Law

is applied to two separate facts in the fields of optics

and linguistics, both discovered by Hermann Grassmann.

He also did advanced work in crystallography, electricity, botany,

folklore, and also wrote on political subjects.

He had little formal training in mathematics, yet single-handedly developed

linear algebra, vector and tensor calculus, multi-dimensional geometry,

new results about cubic surfaces,

the theory of extension, and exterior algebra;

most of this work was so innovative it was not

properly appreciated in his own lifetime.

(Heaviside rediscovered vector analysis many years later.)

Grassmann's exterior algebra, and the associated

concept of Grassmannian manifold, provide a simplifying framework

for many algebraic calculations.

Recently their use led to an important simplification

in quantum physics calculations.Of his linear algebra, one historian wrote "few have come closer than

Hermann Grassmann to creating, single-handedly, a new subject."

Important mathematicians inspired directly by Grassmann

include Peano, Klein, Cartan, Hankel, Clifford, and Whitehead.

__Joseph Liouville__

Liouville did expert research in several areas

including number theory, differential geometry, complex

analysis (especially Sturm-Liouville theory, boundary value problems

and dynamical analysis),

harmonic functions,

topology and mathematical physics.

Several theorems bear his name, including

the key result that any boundedtout

function must be constant (the Fundamental Theorem of

Algebra is an easy corollary of this!);

important results in differential equations,

differential algebra, differential geometry;

a key result about conformal mappings;

and an invariance law about trajectories in phase space

which leads to the Second Law of Thermodynamics and is

key to Hamilton's work in physics.

He was first to prove the existence of transcendental numbers.

(His proof was constructive, unlike that of Cantor which came 30 years later).

He invented Liouville integrability and fractional calculus;

he found a new proof of the Law of Quadratic Reciprocity.

In addition to multiple Liouville Theorems, there are two

"Liouville Principles": a fundamental result in differential algebra,

and a fruitful theorem in number theory.

Liouville was hugely prolific in number theory

but this work is largely overlooked, e.g.

the following remarkable generalization

of Aryabhata's identity:

for all N,

Σ (d_{un}^{3})

= (Σ d_{un})^{2}

oùréis the number of divisors of_{un}un,

and the sums are taken over all divisorsundeN.Liouville established an important journal;

influenced Catalan, Jordan, Chebyshev, Hermite;

and helped promote other mathematicians' work,

especially that of Évariste Galois, whose important results

were almost unknown until Liouville clarified them.

In 1851 Augustin Cauchy was bypassed

to give a prestigious professorship to Liouville instead.

__Ernst Eduard Kummer__

Despite poverty, Kummer became an important mathematician

at an early age, doing work with hypergeometric series,

functions and equations, and number theory.

He worked on the 4-degreeKummer Surfaceet

important algebraic form which inspired Klein's early work.

He solved the ancient problem of finding all rational quadrilaterals.

His most important discovery wasideal numbers;

this led to the theory of ideals and p-adic numbers; cette

discovery's revolutionary nature

has been compared to that of non-Euclidean geometry.

Kummer is famous for his attempts to prove, with the aid

of his ideal numbers, Fermat's Last Theorem.

He established that theorem for almost all exponents (including

all less than 100) but not the general case.Kummer was an inspirational teacher;

his famous students include Cantor, Frobenius, Fuchs, Schwarz,

Gordan, Joachimsthal, Bachmann, and Kronecker.

(Leopold Kronecker was a brilliant genius sometimes ranked ahead of Kummer

in lists like this; that Kummer was Kronecker's teacher at

high school persuades me to give Kummer priority.)

__Évariste Galois__

Galois, who died before the age of twenty-one, not only never

became a professor, but was barely allowed to study as

an undergraduate.

His output of papers, mostly published posthumously,

is much smaller than most of the others on this list, yet it is

considered among the most awesome works in mathematics.

He applied group theory to the theory of equations,

revolutionizing both fields.

(Galois coined the mathematical termgruppe.)

While Abel was the first to prove that some polynomial

equations had no algebraic solutions, Galois established

the necessary and sufficient condition for algebraic solutions to exist.

His principal treatise was a letter he wrote

the night before his fatal duel, of which

Hermann Weyl wrote: "This letter, if judged by the novelty and profundity

of ideas it contains, is perhaps the most substantial piece of writing in

the whole literature of mankind."Galois' ideas were very far-reaching; for example he is

credited as first to prove that trisecting a general angle with

Plato's rules is impossible.

Galois is sometimes cited (instead of Archimedes, Gauss or

Ramanujan) as "the greatest mathematical genius ever."

But he was too far ahead of his time — the top mathematicians of

his day rejected his theory as "incomprehensible."

Galois was persecuted for his Republican politics, imprisoned,

and forced to fight a duel, where he was left to bleed out

without medical attention.

His last words (spoken to his brother) were

"Ne pleure pas, Alfred!

J'ai besoin de tout mon courage pour mourir à vingt ans!"

This tormented life, with its pointless early end, is one

of the great tragedies of mathematical history.

Although Galois' group theory is considered

one of the greatest developments of 19th century mathematics,

Galois' writings were largely ignored until

the revolutionary work of Klein and Lie.

__James Joseph Sylvester__

Sylvester made important contributions in

matrix theory, invariant theory,

number theory, partition theory, reciprocant theory,

geometry, and combinatorics.

He invented the theory of elementary divisors, and co-invented

the law of quadratic forms.

It is said he coined more new mathematical terms (e.g.matrix,

invariant,discriminant,covariant,syzygy,

kurve,Jacobian)

than anyone except Leibniz.

Sylvester was especially noted for the broad range of his mathematics

and his ingenious methods.

He solved (or partially solved) a huge variety of rich puzzles

including various geometric gems;

the enumeration of polynomial roots first tackled by

Descartes and Newton; and, by advancing the theory of

partitions, the system of equations

posed by Euler asThe Problem of the Virgins.

Sylvester was also a linguist, a poet, and did work in mechanics (inventing

the skew pantograph) and optics.

He once wrote, "May not music be described as the mathematics of

the sense, mathematics as music of the reason?"

__Karl Wilhelm Theodor Weierstrass__

Weierstrass devised new definitions for

the primitives of calculus, developed the concept

of uniform convergence, and was

then able to prove several fundamental but hitherto

unproven theorems.

Starting strictly from the integers,

he also applied his axiomatic methods to a definition

of irrational numbers.

He developed important new insights in other fields

including the calculus of variations, elliptic functions, and trigonometry.

Weierstrass shocked his colleagues when he demonstrated

a continuous function which is differentiable nowhere.

(Both this and the Bolzano-Weierstrass Theorem were rediscoveries

of forgotten results by the under-published Bolzano.)

He found simpler proofs of many existing theorems, including

Gauss' Fundamental Theorem of Algebra and the fundamental Hermite-Lindemann

Transcendence Theorem.

Steiner's proof of the Isoperimetric Theorem contained a flaw,

so Weierstrass became the first to supply a fully rigorous

proof of that famous and ancient result.

Peter Dirichlet was a champion of rigor, but Weierstrass

discovered a flaw in the argument for

Dirichlet's Principle of of variational calculus.Weierstrass demonstrated extreme brilliance as a youth,

but during his college years he detoured into drinking and dueling

and ended up as a degreeless secondary school teacher.

During this time he studied Abel's papers, developed results

in elliptic and Abelian functions, proved

the Laurent expansion theorem before Laurent did, and

independently proved the Fundamental Theorem of Functions of

a Complex Variable.

He was interested in power series and felt that others

had overlooked the importance of Abel's Theorem.

Eventually one of his papers was published in a journal;

he was immediately given an honorary doctorate

and was soon regarded as one of the best and most

inspirational mathematicians in the world.

His insistence on absolutely rigorous proofs equaled

or exceeded even that of Cauchy, Abel and Dirichlet.

His students included Kovalevskaya, Frobenius, Mittag-Leffler,

and several other famous mathematicians.

Bell called him "probably the greatest mathematical teacher of all time."

In 1873 Hermite called Weierstrass "the Master of all of us."

Today he is often called the "Father of Modern Analysis."Weierstrass once wrote:

"A mathematician who is not also

something of a poet will never be a complete mathematician."

__George Boole__

George Boole was a precocious child who impressed

by teaching himself classical languages, but was too poor

to attend college and

became an elementary school teacher at age 16.

He gradually developed his math skills; as a young man

he published a paper on the calculus of variations, and soon

became one of the most respected mathematicians in England despite

having no formal training.

He was noted for work in symbolic logic, algebra and analysis,

og også

was apparently the first to discover invariant theory.

When he followed up Augustus de Morgan's earlier work in symbolic

logic, de Morgan insisted that Boole was the true master of that field,

and begged his friend to finally study mathematics at university.

Boole couldn't afford to, and had to be appointed Professor instead!Although very few recognized its importance at the time,

it is Boole's work in Boolean algebra and symbolic logic

for which he is now remembered; this work inspired

computer scientists like Claude Shannon.

Boole's bookAn Investigation of the Laws of Thought

prompted Bertrand Russell to label him

the "discoverer of pure mathematics."Boole once said "No matter how correct a mathematical theorem

may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful."

__Pafnuti Lvovich Chebyshev__

Pafnuti Chebyshev (Pafnuty Tschebyscheff)

was noted for work in probability, number theory,

approximation theory, integrals, the theory of equations,

and orthogonal polynomials.

His famous theorems cover a diverse range;

they include a new version of the Law of Large Numbers,

first rigorous proof of the Central Limit Theorem,

and an important result in integration of radicals first conjectured by Abel.

He invented the Chebyshev polynomials, which have very wide application;

many other theorems or concepts are also named after him.

He did very important work with prime numbers,

proving that there is always a prime

between anynet2n,

and working with the zeta function before Riemann did.

He made much progress with the Prime Number Theorem,

provingdeuxdistinct forms of that theorem,

each incomplete but in a different way.

Chebyshev was very influential for Russian mathematics,

inspiring Andrei Markov and Aleksandr Lyapunov among others.Chebyshev was also a premier applied mathematician and a renowned inventor;

his several inventions include the Chebyshev linkage,

a mechanical device to convert rotational motion to straight-line motion.

He once wrote "To isolate mathematics from the practical demands of

the sciences is to invite the sterility of a cow shut away from the bulls."

__Arthur Cayley__

Cayley was one of the most prolific

mathematicians in history;

a list of the branches of mathematics he pioneered will

seem like an exaggeration.

In addition to being very inventive,

he was an excellent algorist; some considered him

to be the greatest mathematician of the late 19th century

(an era that includes Weierstrass and Poincaré).

Cayley was the essential founder of modern group theory,

matrix algebra, the theory of higher singularities,

and higher-dimensional geometry

(building on Plücker's work and anticipating the ideas of Klein),

as well as the theory of invariants.

Among his many important theorems are

the Cayley-Hamilton Theorem,

and Cayley's Theorem itself

(that any group is isomorphic to a subgroup of a symmetric group).

He extended Hamilton's quaternions and developed the octonions,

but was still one of the first to realize that these special

algebras could often be subsumed by general matrix methods.

(Hamilton's friend John T. Graves independently discovered the

octonions about the same time as Cayley did.)

He also did original research in combinatorics (e.g. enumeration of trees),

elliptic and Abelian functions,

and projective geometry.

One of his famous geometric theorems is a generalization of

Pascal's Mystic Hexagram result; another resulted in an elegant

proof of the Quadratic Reciprocity law.Cayley may have been the

du moinseksentriske

of the great mathematicians:

In addition to

his life-long love of mathematics, he enjoyed hiking,

painting, reading fiction, and had a happy married life.

He easily won Smith's Prize and Senior Wrangler at Cambridge,

but then worked as a lawyer for many years.

He later became professor,

and finished his career in the limelight as

President of the British Association for the Advancement of Science.

He and James Joseph Sylvester

were a source of inspiration to each other.

These two, along with Charles Hermite, are considered the

founders of the important theory of invariants.

Though applied first to algebra, the notion of invariants

is useful in many areas of mathematics.Cayley once wrote:

"As for everything else, so for a mathematical theory: beauty can be

perceived but not explained."

__Charles Hermite__

Hermite studied the works of Lagrange

and Gauss from an early age and soon developed an alternate

proof of Abel's famous quintic impossibility result.

He attended the same college as Galois and

also had trouble passing their examinations, but soon became

highly respected by Europe's best mathematicians for his

significant advances in analytic number theory, elliptic functions,

and quadratic forms.

Along with Cayley and Sylvester,

he founded the important theory of invariants.

Hermite's theory of transformation allowed him to connect

analysis, algebra and number theory in novel ways.

He was a kindly modest man and an inspirational teacher.

Among his students was Poincaré, who said of Hermite, "He never

evokes a concrete image, yet you soon perceive that the more

abstract entities are to him like living creatures….

Methods always seemed to be born in his mind in some mysterious way."

Hermite's other famous students included Darboux, Borel, and

Hadamard who wrote of "how magnificent Hermite's teaching was,

overflowing with enthusiasm for science, which seemed to come to life

in his voice and whose beauty he never failed to communicate to us,

since he felt it so much himself to the very depth of his being."Although he and Abel had proved that the general quintic lacked

algebraic solutions, Hermite introduced an elliptic analog to the

circular trigonometric functions and used these to provide

a general solution for the quintic equation.

He developed the concept of complex conjugate which is now

ubiquitous in mathematical physics and matrix theory.

He was first to prove that the Stirling

and Euler generalizations of the factorial function are equivalent.

He was first to note remarkable facts about Heegner numbers, e.g.

e^{π√163}= 262537412640768743.9999999999992…

(Without computers he was able to calculate this number,

including the twelve 9's to the right of the decimal point.)

Very many elegant concepts and theorems are named after Hermite.

Hermite's most famous result may be his intricate proof

que(along with a broad classe

of related numbers) is transcendental.

(Extending the proof toπwas left to Lindemann,

a matter of regret for historians, some of whom who regard Hermite as the

greatest mathematician of his era.)

__Ferdinand Gotthold Max Eisenstein__

Eisenstein was born into severe poverty and suffered health

problems throughout his short life, but was still one of the more significant

mathematicians of his era.

Today's mathematicians who study Eisenstein are invariably amazed

by his brilliance and originality.

He made revolutionary advances in number theory,

algebra and analysis, and was also a composer of music.

He anticipated ring theory, developed a new basis for elliptic

functions, studied ternary quadratic forms,

proved several theorems about

cubic and higher-degree reciprocity, discovered the

notion of analytic covariant, and much more.Eisenstein was a young prodigy; he once wrote

"As a boy of six I could understand the proof of a mathematical theorem

more readily than that meat had to be cut with one's knife, not one's fork."

Despite his early death, he is considered one

of the greatest number theorists ever.

Gauss named Eisenstein, along with Newton and Archimedes, as one of

the three epoch-making mathematicians of history.

__Leopold Kronecker__

Kronecker was a businessman who pursued mathematics mainly

as a hobby, but was still very prolific,

and one of the greatest theorem provers of his era.

He explored a wide variety of mathematics — number theory,

algebra, analysis, matrixes — and especially the interconnections between

areas.

Many concepts and theorems are named after Kronecker;

some of his theorems are frequently used as lemmas in algebraic number theory,

ergodic theory, and approximation theory.

He provided key ideas about foundations and continuity

despite that he had philosophic objections to

irrational numbers and infinities.

He also introduced the Theory of Divisors to avoid Dedekind's Ideals;

the importance of this and other work was only realized long after

his death.

Kronecker's philosophy eventually led to the Constructivism

and Intuitionism of Brouwer and Poincaré.

__Georg Friedrich Bernhard Riemann__

Riemann was a phenomenal genius whose work

was exceptionally deep, creative and rigorous; Han lagde

revolutionary contributions in many areas of pure mathematics,

and also inspired the development of physics.

He had poor physical health and died at an early age, yet is still

considered to be among the most productive mathematicians ever.

He made revolutionary advances in complex analysis, which he connected to

both topology and number theory.

He applied topology to analysis, and analysis to number theory,

making revolutionary contributions to all three fields.

He introduced the Riemann integral which clarified analysis.

He developed the theory ofmanifolds, a term which he invented.

Manifolds underpin topology.

By imposing metrics on manifolds Riemann invented

differential geometry and took non-Euclidean geometry far

beyond his predecessors.

Riemann's other masterpieces include tensor analysis,

the theory of functions, and a key relationship between

some differential equation solutions and hypergeometric series.

His generalized notions of distance and curvature

described new possibilities for the geometry of space itself.

Several important theorems and concepts are named

after Riemann, e.g. the Riemann-Roch Theorem, a key connection

among topology, complex analysis and algebraic geometry.

He proved Riemann's Rearrangement Theorem, a strong (and paradoxical)

result about conditionally convergent series.

He was also first to prove theorems named after others, e.g. Green's Theorem.

He was so prolific and original that some of his work

went unnoticed (for example, Weierstrass became famous for showing

a nowhere-differentiable continuous function;

later it was found that Riemann had casually mentioned one

in a lecture years earlier).

Like his mathematical peers (Gauss, Archimedes, Newton), Riemann

was intensely interested in physics.

His theory unifying electricity, magnetism and light was

supplanted by Maxwell's theory; however modern physics, beginning

with Einstein's relativity, relies on Riemann's

curvature tensor and other notions of the geometry of space.Riemann's teacher was Carl Gauss, who helped steer the young

genius towards pure mathematics. Gauss selected "On the hypotheses

that Lie at the Foundations of Geometry" as Riemann's first lecture;

with this famous lecture Riemann went far beyond Gauss' initial effort

in differential geometry, extended it to multiple dimensions, and

introduced the new and important theory of differential manifolds.

Five years later, to celebrate his election to the Berlin Academy,

Riemann presented a lecture "On the Number of Prime Numbers Less

Than a Given Quantity," for which "Number" he presented and

partially proved anexactementformula, albeit weirdly complicated.

Numerous papers have been written on the distribution of primes,

but Riemann's contribution is incomparable, despite that

his Berlin Academy lecture was his only paper ever on the topic,

and number theory was far from his specialty.

In the lecture he posed theHypothesis of Riemann's zeta function,

needed for the missing step in his proof.

This Hypothesis is considered the most important and famous unsolved problem

in mathematics.

(Asked what he would first do, if he were magically awakened after

centuries, David Hilbert replied "I would ask whether

anyone had proved the Riemann Hypothesis.")

ζ(.)was defined for convergent cases in Euler's mini-bio,

which Riemann extended via analytic continuation for all cases.

The Riemann Hypothesis "simply" states that

in all solutions ofζ(s = a+b,Je) = 0

non plusshas real parta=1/2

or imaginary partb=0.Despite his great creativity (Gauss praised

Riemann's "gloriously fertile originality;" another biographer called

him "one of the most profound and imaginative mathematicians of all time

(and) a great philosopher"),

Riemann once said: "If only I had the theorems!

Then I should find the proofs easily enough."

__Henry John Stephen forgeron__

Henry Smith (born in Ireland) was one of the greatest number theorists,

working especially with elementary divisors; he also advanced

the theory of quadratic forms.

A famous problem of Eisenstein was, givennet

, in how many different ways canknbe expressed as the sum of

squares?k

Smith made great progress on this problem, subsuming special

cases which had earlier been famous theorems.

Although most noted for number theory, he had great breadth.

He did prize-winning work in geometry, discovered the unique normal form

for matrices which now bears his name, anticipated specific fractals

including the Cantor set, the Sierpinski gasket and the Koch snowflake,

and wrote a paper demonstrating the limitations of Riemann integration.Smith is sometimes called "the mathematician the world forgot."

His paper on integration could have led directly

to measure theory and Lebesgue integration, but was ignored for decades.

The fractals he discovered are named after people who rediscovered them.

The Smith-Minkowski-Siegel mass formula of lattice theory would be

called just the Smith formula, but had to be rediscovered.

And his solution to the Eisenstein five-squares problem, buried

in his voluminous writings on number theory, was ignored:

this "unsolved" problem was featured for a prize which Minkowski won

two decades later!Henry Smith was an outstanding intellect with a modest

and charming personality.

He was knowledgeable in a broad range of fields unrelated

to mathematics; his University even insisted he run for Parliament.

His love of mathematics didn't depend on utility: he once wrote

"Pure mathematics: may it never be of any use to anyone."

__Antonio Luigi Gaudenzio Giuseppe Cremona__

Luigi Cremona made many important advances in analytic,

synthetic and projective geometry, especially in the transformations

of algebraic curves and surfaces.

Working in mathematical physics, he developed the new field of

graphical statics, and used it to reinterpret some of Maxwell's results.

He improved (or found brilliant proofs for) several results of Steiner,

especially in the field of cubic surfaces.

(Some of this work was done in collaboration with Rudolf Sturm.)

He is especially noted for developing the theory of Cremona transformations

which have very wide application.

He found a generalization of Pascal's Mystic Hexagram.

Cremona also played a political role in establishing the modern Italian state

and, as an excellent teacher, helped make Italy a top

center of mathematics.

__James Clerk Maxwell__

At the age of 14, Maxwell published a remarkable paper

on the construction of ovals;

these were an independent discovery of theOvals of Descartes,

but Maxwell allowed more than two foci, had elaborate configurations

(he was drawing the ovals with string and pencil), and identified

errors in Descartes' treatment of them.

His genius was soon renowned throughout Scotland, with the future Lord Kelvin

remarking that Maxwell's "lively imagination started so many hares that

before he had run one down he was off on another."

He did a comprehensive analysis of Saturn's rings; utviklet

the important kinetic theory of gases; explored elasticity, viscosity,

knot theory, topology, soap bubbles, and more.

He introduced the "Maxwell's Demon" as a thought experiment for thermodynamics;

his paper "On Governors" effectively founded the field of cybernetics;

he advanced the theory of color, and produced the first color photograph.

One Professor said of him, "there is scarcely a single topic that

he touched upon, which he did not change almost beyond recognition."

Maxwell was also a poet.Maxwell did little of importance in pure mathematics, so

his great creativity in mathematical physics might not seem enough to

qualify him for this list,

although his contribution to the kinetic theory of gases

(which even led to the first estimate of molecular sizes) would

already be enough to make him one of the greatest physicists.

But then, in 1864 James Clerk Maxwell stunned the world by

publishing the equations of electricity and magnetism,

predicting the existence of radio waves and that light itself

is a form of such waves and is thus linked to the electro-magnetic force.

Richard Feynman considered this the most significant event of

the 19th century (though others might give higher billing to

Darwin's theory of evolution).

Along with Einstein, Newton, Galileo and Archimedes,

Maxwell would be a frequent choice for a Five Greatest Scientists list.

Recalling Newton's comment about "standing on the shoulders" of

earlier greats, Einstein was asked whose shouldershan

stood on; he didn't name Newton: he said "Maxwell."

Maxwell has been called the "Father of Modern Physics"; he ranks #24 on

Hart's list of the Most Influential Persons in History.

__Julius Wilhelm Richard Dedekind__

Dedekind was one of the most innovative mathematicians ever;

his clear expositions and rigorous axiomatic methods had great influence.

He made seminal contributions to abstract algebra and algebraic number theory

as well as mathematical foundations.

He was one of the first to pursue Galois Theory, making major advances there

and pioneering in the application of group theory to other branches

of mathematics.

Dedekind also invented a system of fundamental axioms for arithmetic,

worked in probability theory and complex analysis,

and invented prime partitions and modular lattices.

Dedekind may be most famous for his theory of ideals and rings;

Kronecker and Kummer had begun this, but

Dedekind gave it a more abstract and productive basis,

which was developed further by Hilbert, Noether and Weil.

Though the termringeitself was coined by Hilbert,

Dedekind introduced the termsmodule,feltetideell.

Dedekind was far ahead of his time, so Noether became famous

as the creator of modern algebra; but she acknowledged her great predecessor,

frequently saying "It is all already in Dedekind."Dedekind was concerned with rigor, writing

"nothing capable of proof ought to be accepted without proof."

Before him, the real numbers, continuity, and infinity all

lacked rigorous definitions.

The axioms Dedekind invented allow the integers and rational

numbers to be built and hisDedekind Cutthen led to a rigorous

and useful definition of the real numbers.

Dedekind was a key mentor for Georg Cantor:

he introduced the notion that a bijection implied equinumerosity,

used this to define infinitude (a set is infinite if

equinumerous with its proper subset),

and was first to prove the Cantor-Bernstein-Schröder Theorem,

though he didn't publish his proof.

(Because he spent his career at a minor university,

and neglected to publish some of his work,

Dedekind's contributions may be underestimated.)

__Rudolf Friedrich Alfred Clebsch__

Alfred Clebsch began in mathematical physics,

working in hydrodynamics and elasticity, but went on

to become a pure mathematician of great brilliance and versatility.

He started with novel results in analysis, but went on to

make important advances to the invariant theory of

Cayley and Sylvester (and Salmon and Aronhold),

to the algebraic geometry and elliptic functions of Abel and Jacobi,

and to the enumerative and projective geometries of Plücker.

He was also one of the first to build on Riemann's innovations.

Clebsch developed new notions, e.g. Clebsch-Aronhold symbolic notation

and 'connex';

and proved key theorems about cubic surfaces

(for example, the Sylvester pentahedron conjecture)

and other high-degree curves,

and representations (bijections) between surfaces.

Some of his work, e.g. Clebsch-Gordan coefficients which are important

in physics, was done in collaboration with Paul Gordan.

For a while Clebsch was one of the top mathematicians in Germany,

and founded an important journal, but he died young.

He was a key teacher of Max Noether, Ferdinand Lindemann,

Alexander Brill and Gottlob Frege.

Clebsch's great influence is suggested by the fact that

his name appeared as co-author on a text published 60 years after his death.

__Eugenio Beltrami__

Beltrami was an outstanding mathematician noted

for early insights connecting geometry and topology (differential

geometry, pseudospherical surfaces, etc.), transformation theory,

differential calculus, and especially for proving

the equiconsistency of hyperbolic and Euclidean geometry

for every dimensionality;

he achieved this by building on models of Cayley, Klein,

Riemann and Liouville.

He was first to invent singular value decompositions.

(Camille Jordan and J.J. Sylvester each invented it independently

a few years later.)

Using insights from non-Euclidean geometry, he did important

mathematical work in a very wide range of physics;

for example he improved Green's theorem, generalized the Laplace operator,

studied gravitation in non-Euclidean space,

and gave a new derivation of Maxwell's equations.

__Marie Ennemond Camille Jordanie__

Jordan was a great "universal mathematician",

making revolutionary advances in group theory, topology, and

operator theory;

and also doing important work in differential equations, number theory,

measure theory,

matrix theory, combinatorics, algebra and especially Galois theory.

He worked as both mechanical engineer and professor of analysis.

Jordan is especially famous for the Jordan Closed Curve Theorem of topology,

a simple statement "obviously true" yet remarkably difficult to prove.

In measure theory he developed Peano-Jordan "content"

and proved the Jordan Decomposition Theorem.

He also proved the Jordan-Holder Theorem of group theory,

invented the notion of homotopy,

invented the Jordan Canonical Forms of matrix theory,

and supplied the first complete proof of

Euler's Polyhedral Theorem,F+V = E+2.

Some consider Jordan second only to Weierstrass among great

19th-century teachers; his work inspired such mathematicians

as Klein, Lie and Borel.

__Marius Sophus Lie__

Lie was twenty-five years old before his interest in

and aptitude for mathematics became clear,

but then did revolutionary

work with continuous symmetry and continuous transformation groups.

These groups and the algebra he developed to manipulate

them now bear his name; they have major importance in

the study of differential equations.

Lie sphere geometry is one result of Lie's fertile approach

and even led to a new approach for Apollonius'

ancient problem about tangent circles.

Lie became a close friend and collaborator of Felix Klein early in

their careers; their methods of relating group theory to

geometry were quite similar;

but they eventually fell out after Klein became (unfairly?)

recognized as the superior of the two.

Lie's work wasn't properly appreciated in his own lifetime, but

one later commentator was "overwhelmed by the richness and beauty

of the geometric ideas flowing from Lie's work."

__Jean Gaston Darboux__

Darboux did outstanding work in geometry,

differential geometry, analysis, function theory,

mathematical physics, and other fields,

his ability "based on a rare combination of

geometrical fancy and analytical power."

He devised the Darboux integral, equivalent to Riemann's integral

but simpler;

developed a novel mapping between (hyper-)sphere

and (hyper-)plane; proved an important Envelope Theorem in the

calculus of variations; developed the field

of infinitesimal geometry; et plus

Several important theorems are named after him including

a generalization of Taylor series, the foundational theorem

of symplectic geometry, and the fact that "the image of an interval

is also an interval."

He wrote the definitive textbook on differential geometry;

he was an excellent teacher, inspiring Borel, Cartan and others.

__William Kingdon Clifford__

Clifford was a versatile and talented mathematician who was

among the first to appreciate the work of both Riemann and Grassmann.

He found new connections between algebra, topology and non-Euclidean geometry.

Combining Hamilton's quaternions, Grassmann's exterior algebra,

and his own geometric intuition and understanding of physics,

he developed biquaternions, and generalized this

àgeometric algebra, which paralleled work by Klein.

In addition to developing theories, he also produced ingenious proofs;

for example he was first to prove Miquel's n-Circle Theorem,

and did so with a purely geometric argument.

Clifford is especially famous for anticipating, before Einstein,

that gravitation could be modeled with a non-Euclidean space.

He was a polymath; a talented teacher, noted philosopher,

writer of children's fairy tales, and outstanding athlete.

With his singular genius, Clifford

would probably have become one of the greatest mathematicians

of his era had he not died at age thirty-three.

__Georg Cantor__

Cantor did brilliant and important work early in his career,

par exemple

he greatly advanced the Fourier-series uniqueness question which

had intrigued Riemann. In his explorations of that problem he was led

to questions of set enumeration, and his greatest invention: set theory.

Cantor created modern Set Theory almost single-handedly,

defining cardinal numbers, well-ordering, ordinal numbers,

and discovering the Theory of Transfinite Numbers.

He defined equality between cardinal numbers based on the

existence of a bijection, and was the first to demonstrate that

the real numbers have a higher cardinal number than the

integers.

(He also showed that the rationals have the same cardinality

as the integers; and that the reals have the same cardinality

as the points of N-space and as the power-set of the integers.)

Although there are infinitely many distinct transfinite numbers,

Cantor conjectured thatC, the cardinality of

the reals, was the second smallest transfinite number.

cetteContinuum Hypothesiswas included in Hilbert's famous

List of Problems, and was partly resolved many years later:

Cantor's Continuum Hypothesis is an "Undecidable Statement"

of Set Theory.Cantor's revolutionary set theory attracted vehement opposition

from Poincaré ("grave disease"), Kronecker (Cantor was a "charlatan"

and "corrupter of youth"), Wittgenstein ("laughable nonsense"),

and even theologians.

David Hilbert had kinder words for it:

"The finest product of mathematical genius and one of the

supreme achievements of purely intellectual human activity"

and addressed the critics with "no one shall expel us from

the paradise that Cantor has created."

Cantor's own attitude was expressed with

"The essence of mathematics lies in its freedom."

Cantor's set theory laid the theoretical basis for the measure theory

developed by Borel and Lebesgue.

Cantor's invention of modern set theory

is now considered one of the most important and creative achievements

in modern mathematics.Cantor demonstrated much breadth

(he even involved himself in the Shakespeare authorship controversy!).

In addition to his set theory and key discoveries in the

theory of trigonometric series, he made advances in number theory,

and gave the modern definition of irrational numbers.

HisCantor setwas the early inspiration for fractals.

Cantor was also an excellent violinist.

He once wrote

"In mathematics the art of proposing a question must be held

of higher value than solving it."

__Friedrich Ludwig Gottlob Frege__

Gottlob Frege developed the first complete and fully rigorous

system of pure logic;

his work has been called the greatest advance in logic since Aristotle.

He introduced the essential notion ofquantifiers; he distinguished

terms from predicates, and simple predicates from 2nd-level predicates.

From his second-order logic he defined numbers, and derived

the axioms of arithmetic with what is now called Frege's Theorem.

His work was largely underappreciated at the time, partly

because of his clumsy notation, partly because his system

was published with a flaw (Russell's antinomy).

(Bertrand Russell reports that when he informed him of this flaw, Frege

took it with incomparable integrity, grace, and even intellectual pleasure.)

Frege and Cantor were the era's outstanding foundational theorists;

unfortunately their relationship with each other became bitter.

Despite all this, Frege's work influenced Peano, Russell, Wittgenstein

and others; and he is now often called the greatest

mathematical logician ever.Frege also did work in geometry and differential equations;

and, in order to construct the real numbers with his set theory,

proved an important new theorem of group theory.

He was also an important philosopher, and wrote "Every good mathematician

is at least half a philosopher, and every good philosopher is at least

half a mathematician."

__Ferdinand Georg Frobenius__

Frobenius did significant work in a very broad range of mathematics,

was an outstanding algorist,

and had several successful students including Edmund Landau, Issai Schur,

and Carl Siegel.

In addition to developing the theory of abstract groups,

Frobenius did important work in number theory, differential equations,

elliptic functions, biquadratic forms, matrixes, and algebra.

He was first to actually prove the important Cayley-Hamilton Theorem,

and first to extend the Sylow Theorems to abstract groups.

He anticipated the important and imaginative Prime Density Theorem,

though he didn't prove its general case.

Although he modestly left his name off the "Cayley-Hamilton Theorem,"

many lemmas and concepts are named after him, including

Frobenius conjugacy class, Frobenius reciprocity,

Frobenius manifolds, the Frobenius-Schur Indicator, etc.

He is most noted for hischaracter theory,

a revolutionary advance which led to the representation theory of groups,

and has applications in modern physics.

The middle-aged Frobenius invented this after the aging Dedekind asked him

for help in solving a key algebraic factoring problem.

__Christian Felix Klein__

Klein's key contribution was an application of

invariant theory to unify geometry with group theory.

This radical new view of geometry

inspired Sophus Lie's Lie groups, and also led

to the remarkable unification of Euclidean and non-Euclidean geometries

which is probably Klein's most famous result.

Klein did other work in function theory, providing links between

several areas of

mathematics including number theory, group theory,

hyperbolic geometry, and abstract algebra.

His Klein's Quartic curve and popularly-famous

Klein's bottlewere among several useful results

from his new approaches to groups and higher-dimensional geometries

and equations.

Klein did significant work in mathematical physics,

e.g. writing about gyroscopes.

He facilitated David Hilbert's early career, publishing

his controversial Finite Basis Theorem and declaring it

"without doubt the most important work on

general algebra (the leading German journal) ever published."Klein is also famous for his book on the icosahedron,

reasoning from its symmetries to

develop the elliptic modular and automorphic functions which he

used to solve the general quintic equation.

He formulated a "grand uniformization theorem" about automorphic functions

but suffered a health collapse before completing the proof.

His focus then changed to teaching; he devised a mathematics

curriculum for secondary schools which

had world-wide influence.

Klein once wrote "… mathematics has been most advanced by those who

distinguished themselves by intuition rather than by rigorous proofs."

__Oliver Heaviside__

Heaviside dropped out of high school to teach himself

telegraphy and electromagnetism, becoming first a telegraph

operator but eventually perhaps the greatest electrical

engineer ever.

He developed transmission line theory, invented the coaxial cable,

predicted Cherenkov radiation,

described the use of the ionosphere in radio transmission, and much more.

Some of his insights anticipated parts of special relativity, and

he was first to speculate about gravitational waves.

For his revolutionary discoveries in electromagnetism and mathematics,

Heaviside became the first winner of the Faraday Medal.As an applied mathematician, Heaviside developed operational

calculus (an important shortcut for solving differential equations);

developed vector analysis independently of Grassmann; et

demonstrated the usage of complex numbers for electro-magnetic equations.

Four of the famous Maxwell's Equations are in fact due to Oliver Heaviside,

Maxwell having presented a more cumbersome version.

Although one of the greatest applied mathematicians, Heaviside

is omitted from the Top 100 because he didn't provide proofs for his methods.

Of this Heaviside said,

"Should I refuse a good dinner simply because I do not

understand the process of digestion?"

__Sofia Vasilyevna Kovalevskaya__

Sofia Kovalevskaya (aka Sonya Kowalevski;

née Korvin-Krukovskaya)

was initially self-taught, sought out Weierstrass as her teacher,

and was later considered the greatest female mathematician ever

(before Emmy Noether).

She was influential in the development of Russian mathematics.

Kovalevskaya studied Abelian integrals and partial differential equations,

producing the important Cauchy-Kovalevsky Theorem;

her application of complex analysis to physics inspired Poincaré

and others.

Her most famous work was the solution to theKovalevskaya top,

which has been called a "genuine highlight of 19th-century mathematics."

Other than the simplest cases solved by Euler and Lagrange,

exact ("integrable") solutions to the

equations of motion were unknown, so Kovalevskaya received fame

and a rich prize when she solved theKovalevskaya top.

Her ingenious solution might be considered a mere curiosity,

but since it is still the only post-Lagrange physical motion problem for which

an "integrable" solution has been demonstrated, it

remains an important textbook example.

Kovalevskaya once wrote "It is impossible to be a mathematician

without being a poet in soul."

She was also a noted playwright.

__Jules Henri Poincaré__

Poincaré founded the theory of algebraic (combinatorial)

topology, and is sometimes called the "Father of Topology"

(a title also used for Euler and Brouwer).

He also did brilliant work in several other areas of mathematics;

he was one of the most creative mathematicians ever,

and the greatest mathematician of the Constructivist

("intuitionist") style.

He published hundreds of papers on a variety of topics and might have

become the most prolific mathematician ever,

but he died at the height of his powers.

Poincaré was clumsy and absent-minded; like Galois,

he was almost denied admission to French University,

passing only because at age 17 he was already far too famous to flunk.In addition to his topology,

Poincaré laid the foundations of homology;

he discovered automorphic functions (a unifying

foundation for the trigonometric and elliptic functions),

and essentially founded the theory of periodic orbits;

he made major advances in the theory of differential equations.

He is credited with partial solution of Hilbert's 22nd Problem.

Several important results carry his name, for example the

famous Poincaré Recurrence Theorem, which almost seems to

contradict the Second Law of Thermodynamics.

Poincaré is especially noted for effectively discovering chaos theory,

and for posingPoincaré's Conjecture;

that conjecture was one of the most famous

unsolved problems in mathematics for an entire century,

and can be explained without equations to a layman.

The Conjecture is that all "simply-connected" closed manifolds

are topologically equivalent to "spheres"; it is directly

relevant to the possible topology of our universe.

Recently Grigori Perelman proved Poincaré's conjecture,

and is eligible for the first Million Dollar

math prize in history.As were most of the greatest mathematicians,

Poincaré was intensely interested in physics.

He made revolutionary advances in fluid dynamics and

celestial motions; he anticipated Minkowski space

and much of Einstein's Special Theory of Relativity

(including the famous equationE = mc).^{2}

Poincaré also found time to become

a famous popular writer of philosophy, writing,

"Mathematics is the art of giving the same name to different things;"

and "A (worthy) mathematician experiences in his work

the same impression as an artist; hans

pleasure is as great and of the same nature;"

and "If nature were not beautiful, it would not be worth knowing,

and if nature were not worth knowing, life would not be worth living."

With his fame, Poincaré helped the world recognize the importance of the

new physical theories of Einstein and Planck.

__Andrei Andreyevich Markov__

Markov did excellent work in a broad range of mathematics

including analysis, number theory, algebra,

continued fractions, approximation theory, and especially

probability theory:

it has been said that his accuracy and clarity transformed

probability theory into one of the most perfected areas of mathematics.

Markov is best known as the founder of the theory of stochastic processes.

In addition to his Ergodic Theorem about such processes, theorems

named after him include the Gauss-Markov Theorem of statistics,

the Riesz-Markov Theorem of functional analysis, and

the Markov Brothers' Inequality in the theory of equations.

Markov was also noted for his politics, mocking Czarist rule,

and insisting that he be excommunicated from the Russian Orthodox Church

when Tolstoy was.Markov had a son, also named Andrei Andreyevich, who was also an

outstanding mathematician of great breadth.

Among the son's achievements was Markov's Theorem, which helps relate

the theories of braids and knots to each other.

__Giuseppe Peano__

Giuseppe Peano is one of the most under-appreciated

of all great mathematicians.

He started his career by proving a fundamental

theorem in differential equations, developed practical

solution methods for such equations, discovered a continuous

space-filling curve (then thought impossible), and laid

the foundations of abstract operator theory.

He also produced the best calculus textbook of his time,

was first to produce a correct (non-paradoxical) definition of

surface area, proved an important

theorem about Dirichlet functions, did important work

in topology, and much more.

Much of his work was unappreciated and left for others to rediscover:

he anticipated many of Borel's and Lebesgue's

results in measure theory, and several concepts and theorems

of analysis.

He was the champion of counter-examples, and found flaws in

published proofs of several important theorems.Most of the preceding work was done when Peano was quite young.

Later he focused on mathematical foundations, and this is the work

for which he is most famous.

He developed rigorous definitions and axioms for

set theory, as well as most of the notation of modern set theory.

He was first to define arithmetic (and then the rest

of mathematics) in terms of set theory.

Peano was first to note that some proofs required an explicit Axiom of Choice

(although it was Ernst Zermelo who explicitly formulated that Axiom a

few years later).Despite his early show of genius, Peano's quest for utter

rigor may have detracted from his influence in mainstream mathematics.

Moreover, since he modestly referenced work by predecessors like

Dedekind, Peano's huge influence in axiomatic theory is often overlooked.

Yet Bertrand Russell reports that it was from Peano

that he first learned that a single-member set is not the same

as its element; this fact is now taught in elementary school.

__Samuel Giuseppe Vito Volterra__

Vito Volterra founded the field of functional analysis

('functions of lines'), and used it to extend the work of

Hamilton and Jacobi to more areas of mathematical physics.

He developed cylindrical waves and the theory of integral equations.

He worked in mechanics, developed the theory of crystal dislocations,

and was first to propose the use of helium in balloons.

Eventually he turned to mathematical biology and made notable

contributions to that field, e.g. predator-prey equations.

__David Hilbert__

Hilbert, often considered the greatest

mathematician of the 20th century,

was unequaled in many fields of mathematics,

including axiomatic theory, invariant theory,

algebraic number theory, class field theory and functional analysis.

He proved many new theorems, including

the fundamental theorems of algebraic manifolds,

and also discovered simpler proofs for older theorems.

His examination of calculus led him

to the invention ofHilbert space,

considered one of the key concepts of functional analysis

and modern mathematical physics.

His Nullstellensatz Theorem laid the foundation of algebraic geometry.

He was a founder of fields like metamathematics and modern logic.

He was also the founder of the "Formalist" school which opposed the

"Intuitionism" of Kronecker and Brouwer.

He developed a new system of definitions and axioms

for geometry, replacing the 2200 year-old system of Euclid.

As a young Professor he proved his Finite Basis Theorem,

now regarded as one of the most important results of

general algebra.

His mentor, Paul Gordan, had sought the proof for many years, and

rejected Hilbert's proof as non-constructive.

Later, Hilbert produced the first constructive proof of the

Finite Basis Theorem, as well.

In number theory, he proved Waring's famous conjecture

which is now known as the Hilbert-Waring Theorem.Any one man can only do so much, so the

greatest mathematicians should help nurture their

colleagues.

Hilbert provided a famous List of 23 Unsolved

Problems, which inspired and directed the development

of 20th-century mathematics.

Hilbert was warmly regarded by his colleagues and students,

and contributed to the careers of several great mathematicians

and physicists including Georg Cantor, Hermann Minkowski,

John von Neumann, Emmy Noether, Alonzo Church, and Albert Einstein.

His doctoral students included Hermann Weyl, Richard Courant,

Max Dehn, Teiji Takagi, Ernst Zermelo, Wilhelm Ackermann, the chess

champion Emanuel Lasker, and many other famous mathematicians.Eventually Hilbert turned to physics

and made key contributions to classical and quantum

physics and to general relativity.

He published theEinstein Field Equations

independently of Einstein

(though his writings make clear he treats this as strictly

Einstein's invention).

__Hermann Minkowski__

(1864-1909) Lithuania, Germany

Minkowski won a prestigious prize at age 18 for

reconstructing Eisenstein's enumeration of the ways to represent

integers as the sum of five squares.

(The Paris Academy overlooked that Smith had already published

a solution for this!)

His proof built on quadratic forms and continued fractions

and eventually led him to the new field of Geometric Number Theory,

for which Minkowski's Convex Body Theorem (a sort of pigeonhole principle) is

often called the Fundamental Theorem.

Minkowski was also a major figure in the development of functional analysis.

With his "question mark function" and "sausage," he was also a

pioneer in the study of fractals.

Several other important results are named after him, e.g.

the Hasse-Minkowski Theorem.

He was first to extend the Separating Axis Theorem to multiple

dimensions.

Minkowski was one of Einstein's teachers, and also a close friend

of David Hilbert.

He is particularly famous for building on

Poincaré's work to inventMinkowski space

to deal with Einstein's Special Theory of Relativity.

This not only provided a better explanation for the Special Theory,

but helped inspire Einstein toward his General Theory.

Minkowski said

that his "views of space and time … have sprung from the soil

of experimental physics, and therein lies their strength….

Henceforth space by itself, and time by itself, are doomed to fade away

into mere shadows, and only a kind of union of the two will

preserve an independent reality."

__Jacques Salomon Hadamard__

Hadamard made revolutionary advances in several different

areas of mathematics, especially complex analysis, analytic number

theory, differential geometry, partial differential equations,

symbolic dynamics, chaos theory, matrix theory, and Markov chains;

for this reason he

is sometimes called the "Last Universal Mathematician."

He also made contributions to physics.

One of the most famous results in mathematics is

the Prime Number Theorem, that there are approximately

n/log nprimes less thann.

This result was conjectured by Legendre and Gauss,

attacked cleverly by Riemann and Chebyshev,

and finally, by building on Riemann's work,

proved by Hadamard and Vallee-Poussin.

(Hadamard's proof is considered more elegant and useful than Vallee-Poussin's.)

Several other important theorems are named after

Hadamard (e.g. his Inequality of Determinants),

and some of his theorems are named after others

(Hadamard was first to prove Brouwer's Fixed-Point Theorem

for arbitrarily many dimensions).

Hadamard was also influential in promoting others' work:

He is noted for his survey of Poincaré's work;

his staunch defense of the Axiom of Choice led to the acceptance of Zermelo's

travail.

Hadamard was a successful teacher,

with André Weil, Maurice Fréchet, and others acknowledging

him as key inspiration.

Like many great mathematicians he emphasized the importance of intuition, writing

"The object of mathematical rigor is to sanction and

legitimize the conquests of intuition, and there never was any other object for it."

__Felix Hausdorff__

Hausdorff had diverse interests: he composed music

and wrote poetry, studied astronomy, wrote on philosophy,

but eventually focused on mathematics, where he did important work in

several fields including set theory, measure theory,

functional analysis, and both algebraic and point-set topology.

His studies in set theory led him to the

Hausdorff Maximal Principle, and the Generalized Continuum Hypothesis;

his concepts now called Hausdorff measure and Hausdorff dimension led

to geometric measure theory and fractal geometry;

his Hausdorff paradox led directly to the famous Banach-Tarski paradox;

he introduced other seminal concepts, e.g. Hausdorff Distance.

He also worked in analysis, solving the Hausdorff moment problem.As Jews in Hitler's Germany, Hausdorff and his wife committed

suicide rather than submit to internment.

__Élie Joseph Cartan__

Cartan worked in the theory of Lie groups

and Lie algebras,

applying methods of topology, geometry and invariant theory to

Lie theory, and classifying all Lie groups.

This work was so significant that

Cartan, rather than Lie, is considered the most important

developer of the theory of Lie groups.

Using Lie theory and ideas like hisMethod of Prolongation

he advanced the theories of differential equations

and differential geometry.

Cartan introduced several new concepts including

algebraic group, exterior differential forms, spinors,

moving frames, Cartan connections.

He proved several important theorems, e.g. Schläfli's Conjecture

about embedding Riemann metrics, Stokes' Theorem,

and fundamental theorems about symmetric Riemann spaces.

He made a key contribution to Einstein's general relativity,

based on what is now called Riemann-Cartan geometry.

Cartan's methods were so original as to be fully appreciated only recently;

many now consider him to be one of the greatest mathematicians of his era.

In 1938 Weyl called him "the greatest living master in differential geometry."

__Félix Édouard Justin Émile Borel__

Borel exhibited great talent while still in his teens,

soon practically founded modern measure theory, and received

several honors and prizes.

Among his famous theorems is the Heine-Borel Covering Theorem.

He also did important work in several other fields of mathematics,

including divergent series, quasi-analytic functions,

differential equations, number theory, complex analysis,

theory of functions, geometry, probability theory, and game theory.

Relating measure theory to probabilities, he

introduced concepts likenormal numbersand the Borel-Kolmogorov paradox.

He also did work in relativity and the philosophy of science.

He anticipated the concept of chaos, inspiring Poincaré.

Borel combined great creativity with strong analytic power;

however he was especially interested in applications, philosophy, and

education, so didn't pursue the tedium of rigorous development and proof;

for this reason his great importance as a theorist is often underestimated.

Borel was decorated for valor in World War I, entered politics

between the Wars, and joined the French

Resistance during World War II.

__Tullio Levi-Civita__

Levi-Civita was noted for strong geometrical intuition,

and excelled at both pure mathematics and mathematical physics.

He worked in analytic number theory, differential

equations, tensor calculus, hydrodynamics, celestial mechanics,

and the theory of stability.

Several inventions are named after him, e.g.

the non-archimedean Levi-Civita field, the Levi-Civita parallelogramoid,

and the Levi-Civita symbol.

His work inspired all three of the greatest 20th-century mathematical

physicists, laying key mathematical groundwork for Weyl's unified field

theory, Einstein's relativity, and Dirac's quantum theory.

__Henri Léon Lebesgue__

Lebesgue did groundbreaking work in real

analysis, advancing Borel's measure theory;

his Lebesgue integral superseded the Riemann integral

and improved the theoretical basis for Fourier analysis.

Several important theorems are named after

him, e.g. the Lebesgue Differentiation Theorem

and Lebesgue's Number Lemma.

He did important work on Hilbert's 19th Problem, and in

the Jordan Curve Theorem for higher dimensions.

In 1916, the Lebesgue integral was compared "with a modern Krupp gun,

so easily does it penetrate barriers which were impregnable."

In addition to his seminal contributions to measure theory

and Fourier analysis, Lebesgue made significant contributions in

several other fields including complex analysis,

topology, set theory, potential theory, dimension theory,

and calculus of variations.

__Edmund Georg Hermann Landau__

Landau was one of the most prolific

and influential number theorists ever and wrote the first

comprehensive treatment of analytic number theory.

He was also adept at complex function theory.

He was especially keen at finding very simple proofs:

one of his most famous results was a simpler proof

of Hadamard's prime number theorem; being simpler it was also more fruitful

and led to Landau's Prime Ideal Theorem.

In addition to simpler proofs of existing theorems, new theorems by Landau

include important facts about Riemann's Hypothesis;

facts about Dirichlet series;

key lemmas of analysis;

a result in Waring's Problem;

a generalization of the Little Picard Theorem;

and a partial proof of Gauss' conjecture about the density of classes

of composite numbers.

In 1912 Landau described four conjectures about prime numbers which were

'unattackable with present knowledge': (a) Goldbach's conjecture,

(b) infinitely many primes n^2+1, (c) infinitely many twin primes (p, p+2),

(d) a prime exists in every interval (n^2, n^2+n). By 2018 none of these conjectures

have been resolved, though much progress has been made in each case.

Landau was the inventor of big-O notation.

Hardy wrote that no one was ever more passionately devoted to

mathematics than Landau.

__Godfrey Harold Hardy__

Hardy was an extremely prolific research mathematician who

did important work in analysis (especially the theory of integration),

number theory, global analysis, and analytic number theory.

He proved several important theorems about numbers, for example

that Riemann's zeta function has infinitely many zeros

with real part 1/2.

He was also an excellent teacher and wrote several

excellent textbooks, as well as a famous treatise on

the mathematical mind.

He abhorred applied mathematics, treating mathematics as a creative art;

yet his work has found application in

population genetics, cryptography, thermodynamics and

particle physics.Hardy is especially famous (and important)

for his encouragement of and collaboration

with Ramanujan.

Hardy provided rigorous proofs for several of Ramanujan's

conjectures, including Ramanujan's "Master Theorem" of analysis.

Among other results of this collaboration was the

Hardy-Ramanujan Formula for partition enumeration, which

Hardy later used as a model to develop the Hardy-Littlewood

Circle Method;

Hardy then used this method to prove stronger versions

of the Hilbert-Waring Theorem, and in prime number theory;

the method has continued to be a very productive

tool in analytic number theory.

Hardy was also a mentor to Norbert Wiener, another famous prodigy.Hardy once wrote "A mathematician, like a painter or poet,

is a maker of patterns.

If his patterns are more permanent than theirs, it is because

they are made with ideas."

He also wrote "Beauty is the first test; there is no permanent place in

the world for ugly mathematics."

__René Maurice Fréchet__

Maurice Fréchet introduced the concept of metric spaces

(though not using that term); and also made major contributions

to point-set topology.

Building on work of Hadamard and Volterra,

he generalized Banach spaces to use new (non-normed) metrics

and proved that many

important theorems still applied in these more general spaces.

For this work, and his invention of the notion of compactness,

Fréchet is called the Founder of the Theory of Abstract Spaces.

He also did important work in probability theory and in analysis;

for example he proved the Riesz Representation Theorem the same year Riesz did.

Many theorems and inventions are named after him,

for example Fréchet Distance, which has many applications

in applied math, e.g. protein structure analysis.

__Albert Einstein__

(1879-1955) Germany, Switzerland, U.S.A.

Albert Einstein was unquestionably one of the

two greatest physicists in all of history.

The atomic theory achieved general acceptance only after Einstein's

1905 paper, introducing the Einstein-Smoluchowski relation,

showed that atoms' discreteness explained Brownian motion.

Another 1905 paper introduced the famous equation

E = mc; yet Einstein published other papers^{2}

that same year, two of which were more important and influential

than either of the two just mentioned.

No wonder that physicists speak of theMiracle Year

without bothering to qualify it asEinstein's Miracle Year!

(Before his Miracle Year, Einstein had been a mediocre

undergraduate, and held minor jobs including patent

examiner.)

Altogether Einstein published at least 300 books or papers on physics.

For example, in a 1917 paper he anticipated the principle of the laser.

Also, sometimes in collaboration with Leo Szilard, he was co-inventor of

several devices, including a gyroscopic compass,

hearing aid, automatic camera and, most famously,

the Einstein-Szilard refrigerator.

He became a very famous and influential public figure.

(For example, it was his letter that led Roosevelt to start

the Manhattan Project.)

Among his many famous quotations is:

"The search for truth is more precious than its possession."Einstein is most famous for his Special and General Theories

of Relativity, but he should be considered the key pioneer of

Quantum Theory as well, drawing inferences from Planck's work that

no one else dared to draw.

Indeed it was his articulation of the quantum principle in a 1905 paper

which has been called

"the most revolutionary sentence written by a physicist of the

twentieth century."

Einstein's discovery of the photon in that paper led to his only Nobel Prize;

years later, he was first to call attention to the "spooky"

nature of quantum entanglement.

Einstein was also first to call attention to a flaw in Weyl's

earliest unified field theory.

But despite the importance of his other contributions it is Einstein's

General Theory which is his most profound contribution.

Minkowski had developed a flat 4-dimensional space-time to

cope with Einstein's Special Theory; but it was Einstein

who had the vision to add curvature to that space

to describe acceleration.Einstein certainly has the breadth, depth, and historical importance

to qualify for this list; but his genius and significance were not in

the field of pure mathematics.

(He acknowledged his limitation, writing

"I admire the elegance of your (Levi-Civita's) method of computation;

it must be nice to ride through these fields upon the horse of

true mathematics while the like of us have

to make our way laboriously on foot.")

Einsteinétaita mathematician, however;

he pioneered the application of tensor calculus to physics

and invented theEinstein summation notation.

That Einstein's Equation of General Relativity explained a discrepancy

in Mercury's orbit was a discovery made by Einstein personally

(a discovery he described as 'joyous excitement' that gave

him heart palpitations).

He composed a beautiful essay about mathematical proofs

using the Theorem of Menelaus as his example.

The sheer strength and diversity of his intellect is suggested by

his elegant paper on river meanders, a

classic of that field.

Certainly he belongs on a Top 100 List:

his extreme greatness overrides his focus away from math.

Einstein ranks #10 on Michael Hart's famous list of

the Most Influential Persons in History.

His General Theory of Relativity has been called the

most creative and original scientific theory ever.

Einstein once wrote "… the creative principle resides in mathematics

(; thus)

I hold it true that pure thought can grasp reality, as the ancients dreamed."

__Oswald Veblen__

Oswald Veblen's first mathematical achievement was a

novel system of axioms for geometry.

He also worked in topology; projective geometry; differential geometry

(where he was first to introduce the concept of differentiable manifold);

ordinal theory (where he introduced the Veblen hierarchy); et

mathematical physics where he worked with spinors and relativity.

He developed a new theory of ballistics during World War I

and helped plan the first American computer during World War II.

His famous theorems include the Veblen-Young Theorem

(an important algebraic fact about projective spaces);

a proof of the Jordan Curve Theorem more rigorous than Jordan's;

and Veblen's Theorem itself (a generalization

of Euler's result about cycles in graphs).

Veblen, a nephew of the famous economist Thorstein Veblen,

was an important teacher; his famous students included Alonzo Church,

John W. Alexander, Robert L. Moore, and J.H.C. Whitehead.

He was also a key figure in establishing

Princeton's Institute of Advanced Study;

the first five mathematicians he hired for the Institute were

Einstein, von Neumann, Weyl, J.W. Alexander and Marston Morse.

__Luitzen Egbertus Jan Brouwer__

Brouwer is often considered the "Father of Topology;"

among his important theorems were the Fixed Point Theorem,

the "Hairy Ball" Theorem,

the Jordan-Brouwer Separation Theorem,

and the Invariance of Dimension.

He developed the method of simplicial approximations,

important to algebraic topology;

he also did work in geometry, set theory, measure theory,

complex analysis and the foundations of mathematics.

He was first to anticipate forms like theLakes of Wada,

leading eventually to other measure-theory "paradoxes."

Several great mathematicians, including Weyl, were inspired

by Brouwer's work in topology.Brouwer is most famous as the founder of Intuitionism,

a philosophy of mathematics in sharp contrast

to Hilbert's Formalism, but Brouwer's philosophy also

involved ethics and aesthetics and has been compared with

those of Schopenhauer and Nietzsche.

Part of his mathematics thesis was rejected as

"… interwoven with some kind of pessimism and mystical attitude

to life which is not mathematics …"

As a young man, Brouwer spent a few years to develop

topology, but once his great talent was demonstrated

and he was offered prestigious professorships, he devoted

himself to Intuitionism, and acquired a reputation as

eccentric and self-righteous.Intuitionism has had a significant influence,

although few strict adherents.

Since only constructive proofs are permitted, strict adherence

would slow mathematical work.

This didn't worry Brouwer who once wrote:

"The construction itself is an art, its application to the world

an evil parasite."

__Amalie Emmy Noether__

Noether was an innovative researcher

who was considered the greatest master of abstract algebra ever;

her advances included a new theory of ideals, the inverse Galois problem,

and the general theory of commutative rings.

She originated novel reasoning methods, especially one based

on "chain conditions," which advanced invariant theory

and abstract algebra; her insistence on generalization led to a unified

theory of modules and Noetherian rings.

Her approaches tended to unify disparate areas (algebra, geometry,

topology, logic) and led eventually to modern category theory.

Her invention of Betti homology groups led to algebraic topology, and thus

revolutionized topology.Noether's work has found various applications in physics,

and she made direct advances in mathematical physics herself.

Noether's Theorem establishing that certain symmetries imply

conservation laws has been

called the most important Theorem in physics since the Pythagorean Theorem.

Several other important theorems are named after her, e.g.

Noether's Normalization Lemma, which provided an important

new proof of Hilbert's Nullstellensatz.

Noether was an unusual and inspiring teacher; her successful students

included Emil Artin, Max Deuring, Jacob Levitzki, etc.

She was generous with students and colleagues, even allowing them to claim

her work as their own.

Noether was close friends with the other greatest mathematicians of her

generation: Hilbert, von Neumann, and Weyl.

Weyl once said he was embarrassed to accept the famous Professorship

at Göttingen because Noether was his "superior as a mathematician."

Emmy Noether is considered the greatest female mathematician ever.

__Waclaw Sierpinski__

Sierpinski won a gold medal as an undergraduate by

making a major improvement to a famous theorem by Gauss

about lattice points inside a circle.

He went on to do important research in set theory, number theory,

point set topology, the theory of functions, and fractals.

He was extremely prolific, producing 50 books and over 700 papers.

He was a Polish patriot: he contributed to the development of Polish

mathematics despite that his land was controlled by Russians or Nazis for most

of his life. He worked as a code-breaker during the Polish-Soviet War,

helping to break Soviet ciphers.Sierpinski was first to prove Tarski's remarkable conjecture that

the Generalized Continuum Hypothesis implies the Axiom of Choice.

He developed three famous fractals: a space-filling curve;

the Sierpinski gasket; and the Sierpinski carpet, which covers the plane

but has area zero and has found application in antennae design.

Borel had proved that almost all real numbers are "normal" but Sierpinski

was the first to actually display a number which is normal in every base.

He proved the existence of infinitely many Sierpinski numbers having the

property that, e.g.(78557·2est^{n}+1)

a composite number for every natural number n.

It remains an unsolved problem (likely to

be defeated soon with high-speed computers) whether 78557

is the smallest such "Sierpinski number."

__Salomon Lefschetz__

Lefschetz was born in Russia, educated as an engineer

in France, moved to U.S.A., was severely handicapped in an accident,

and then switched to pure mathematics.

He was a key founder of algebraic topology,

even coining the wordtopologi,

and pioneered the application of topology to algebraic geometry.

Starting from Poincaré's work, he developed

Lefschetz duality and used it to derive conclusions about fixed points

in topological mappings.

The Lefschetz Fixed-point Theorem left Brouwer's famous result as just

a special case.

His Picard-Lefschetz theory eventually led to the proof of the Weil conjectures.

Lefschetz also did important work in algebraic geometry,

non-linear differential equations, and control theory.

As a teacher he was noted for a combative style.

Preferring intuition over rigor, he once told a student who had improved on

one of Lefschetz's proofs:

"Don't come to me with your pretty proofs. We don't bother with that

baby stuff around here."

__George David Birkhoff__

Birkhoff is one of the greatest native-born American mathematicians

ever, and did important work in many fields.

There are several significant theorems named after him:

the Birkhoff-Grothendieck Theorem is an important result about vector

bundles;

Birkhoff's Theorem is an important result in algebra;

and Birkhoff's Ergodic Theorem is a key result in statistical mechanics

which has since been applied to many other fields.

His Poincaré-Birkhoff Fixed Point Theorem is especially important

in celestial mechanics, and led to instant worldwide fame:

the great Poincaré had described it as most important,

but had been unable to complete the proof.

In algebraic graph theory, he invented Birkhoff's

chromatic polynomial (while trying to prove the four-color map theorem);

he proved a significant result in general relativity which implied the

existence of black holes;

he also worked in differential equations and number theory;

he authored an important text on dynamical systems.

Like several of the great mathematicians of that era, Birkhoff

developed his own set of axioms for geometry; it is his axioms

that are often found in today's high school texts.

Birkhoff's intellectual interests went beyond mathematics; he once wrote

"The transcendent importance of love and goodwill in all human relations

is shown by their mighty beneficent effect upon the individual and society."

__Hermann Klaus Hugo (Peter) Weyl__

Weyl studied under Hilbert and became one of the premier

mathematicians and thinkers of the 20th century.

Along with Hilbert and Poincaré he was a great "universal"

mathematician; his discovery of gauge invariance and notion of Riemann

surfaces form the basis of modern physics; he was also

a creative thinker in philosophy.

Weyl excelled at many fields of mathematics including integral equations,

harmonic analysis, analytic number theory, Diophantine approximations,

axiomatic theory, and mathematical philosophy;

but he is most respected for his revolutionary advances

in geometric function theory (e.g., differentiable manifolds),

the theory of compact groups (incl. representation theory), and

theoretical physics (e.g., Weyl tensor, gauge field theory and invariance).

His theorems include key lemmas and foundational results in several fields;

Atiyah commented that whenever

he explored a new topic he found that Weyl had preceded him.

Although he was a master of algebra, he revealed his philosophic

preference by

writing "In these days the angel of topology and the devil of abstract

algebra fight for the soul of every individual discipline of mathematics."

For a while, Weyl was a disciple of Brouwer's Intuitionism and

helped advance that doctrine, but he eventually found it too restrictive.

Weyl was also a very influential figure in all three major fields

of 20th-century physics:

relativity, unified field theory and quantum mechanics.

He and Einstein were great admirers of each other.

Because of his contributions to Schrödinger, many think the latter's

famous result should be named the Schrödinger-Weyl Wave Equation.Vladimir Vizgin wrote "To this day, Weyl's (unified field)

theory astounds all in the depth of its ideas, its mathematical simplicity,

and the elegance of its realization."

The Nobel prize-winner Julian Schwinger, himself considered an

inscrutable genius, was so impressed by Weyl's book connecting

quantum physics to group theory that he likened Weyl to a "god"

because "the ways of gods are mysterious, inscrutable, and

beyond the comprehension of ordinary mortals."

Weyl once wrote: "My work always tried to unite the Truth

with the Beautiful, but when I

had to choose one or the other, I usually chose the Beautiful."

__John Edensor Littlewood__

John Littlewood was a very prolific researcher.

(This fact is obscured somewhat in that many papers

were co-authored with Hardy, and their names were always given

in alphabetic order.)

The tremendous span of his career is suggested by the fact

that he won Smith's Prize (and Senior Wrangler) in 1905

and the Copley Medal in 1958.

He specialized in analysis and analytic number theory

but also did important work in combinatorics, Fourier theory,

Diophantine approximations, differential equations, and other fields.

He also did important work in practical engineering, creating a method

for accurate artillery fire during the First World War,

and developing equations for radio and radar in preparation for the Second War.

He worked with the Prime Number Theorem and Riemann's Hypothesis;

and proved the unexpected fact that Chebyshev's bias, and

, while true for most, and all butLi(x)>π(x)

very large, numbers, are violated infinitely often.

(Building on this result, it is now known that there is a big

patch of primes near 10^{9608}that exceed the Li(x)

prediction, though few if any of those primes are actually known.)

Some of his work was elementary, e.g. his elegant proof that a

cube cannot be dissected into unequal cubes; mais

most of his results were too specialized to state here, e.g.

his widely-applied

4/3 Inequalitywhich guarantees that certain bimeasures are finite,

and which inspired one of Grothendieck's most famous results.

Hardy once said that his friend was

"the man most likely to storm and smash a really deep and formidable problem;

there was no one else who could command such a combination of insight,

technique and power."

Littlewood's response was that it was possible to beaussifort

of a mathematician, "forcing

through, where another might be driven to a different,

and possibly more fruitful, approach."

__Srinivasa Ramanujan Iyengar__

Like Abel, Ramanujan was a self-taught prodigy who lived in

a country distant from his mathematical peers, and suffered

from poverty: childhood dysentery and vitamin deficiencies probably led

to his early death.

Yet he produced 4000 theorems or conjectures in number theory,

algebra, and combinatorics.

While some of these were old theorems or just curiosities,

many were brilliant new theorems with very difficult proofs.

For example, he found a beautiful identity connecting

Poisson summation to the Möbius function.

He also found a brilliant generalization of Lagrange's

Four Square Theorem; and much more.

Ramanujan might be almost unknown today, except that

his letter caught the eye of Godfrey Hardy, who saw

remarkable, almost inexplicable formulae which

"must be true, because if they were not true, no one would

have had the imagination to invent them."

Ramanujan's specialties included infinite series,

elliptic functions, continued fractions,

partition enumeration, definite integrals, modular equations,

the divisor function,

gamma functions, "mock theta" functions, hypergeometric series,

and "highly composite" numbers.

Ramanujan's "Master Theorem" has wide application in analysis,

and has been applied to the evaluation of Feynman diagrams.

Much of his best work was done in collaboration with Hardy, for example

a proof that almost all numbersnhave aboutlog log n

prime factors (a result which developed into probabilistic number theory).

Much of his methodology, including unusual ideas about divergent series,

was his own invention.

(As a young man he made the absurd claim that1+2+3+4+… = -1/12.

Later it was noticed that this claim translates to a true statement

about the Riemann zeta function, with which Ramanujan was unfamiliar.)

Ramanujan's innate ability for algebraic manipulations equaled or surpassed

that of Euler and Jacobi.Ramanujan's most famous work was with the partition enumeration

funksjonp(), Hardy guessing that some of these discoveries would have

been delayed at least a century without Ramanujan.

Together, Hardy and Ramanujan developed an analytic approximation to

p(), although Hardy was initially awed by Ramanujan's intuitive

certainty about the existence of such a formula,

and even the form it would have.

(Rademacher and Selberg later discovered an exact

expression to replace the Hardy-Ramanujan approximation;

when Ramanujan's notebooks were studied it

was found he had anticipated their technique, but had

deferred to his friend and mentor.)In a letter from his deathbed, Ramanujan introduced his mysterious

"mock theta functions",

gave examples, and developed their properties.

Much later these forms began to appear in disparate areas:

combinatorics, the proof of Fermat's Last Theorem, and even

knot theory and the theory of black holes.

It was only recently, more than 80 years after Ramanujan's letter,

that his conjectures

about these functions were proven; solutions mathematicians had sought

unsuccessfully were found among his examples.

Mathematicians are baffled that Ramanujan could make these

conjectures, which they confirmed only with difficulty using methods

not available in Ramanujan's day.Many of Ramanujan's results

are so inspirational that there is a periodical dedicated to them.

The theories of strings and crystals have benefited from Ramanujan's work.

(Today some professors achieve fame just by finding

a new proof for one of Ramanujan's many results.)

Unlike Abel, who insisted on rigorous proofs, Ramanujan

often omitted proofs.

(Ramanujan may have had unrecorded proofs, poverty leading him to

use chalk and erasable slate rather than paper.)

Unlike Abel, much of whose work depended on the complex

numbers, most of Ramanujan's work focused on real numbers.

Despite these limitations, some consider Ramanujan to be the

greatest mathematical genius ever; but he ranks as low as #17

since many lesser mathematicians were much more influential.Because of its fast convergence, an odd-looking formula of Ramanujan is

sometimes used to calculateπ:

99^{2}/ π

= √8 ∑_{k=0,∞}

((4k)! (1103+26390 k) / (k!^{4}396^{4k}))

__Thoralf Albert Skolem__

Thoralf Skolem proved fundamental theorems of lattice theory,

proved the Skolem-Noether Theorem of algebra, also worked with set theory

and Diophantine equations; but is best known for his work

in logic, metalogic, and non-standard models.

Some of his work preceded similar results by Gödel.

He developed a theory of recursive functions which anticipated

some computer science.

He worked on the famous Löwenheim-Skolem Theorem which

has the "paradoxical" consequence

that systems with uncountable sets can have countable models.

("Legend has it that Thoralf Skolem, up until the end of his life,

was scandalized by the association of his name to a result of this type,

which he considered an absurdity, nondenumerable sets being, for him,

fictions without real existence.")

__George Pólya__

George Pólya (Pólya György)

did significant work in several fields: complex analysis,

probability, geometry, algebraic number theory, and combinatorics,

but is most noted for his teachingHow to Solve It,

the craft of problem posing and proof.

He is also famous for the Pólya Enumeration Theorem.

Several other important theorems he proved include

the Pólya-Vinogradov Inequality of number theory,

the Pólya-Szego Inequality of functional analysis, and

the Pólya Inequality of measure theory.

He introduced the Hilbert-Pólya Conjecture that the

Riemann Hypothesis might be a consequence of spectral theory

(in 2017 this Conjecture was partially proved by a team of

physicists, and the Riemann Hypothesiskanskjebe close

to solution!).

He introduced the famous "All horses are the same color" example

of inductive fallacy; he named the Central Limit Theorem of statistics.

Pólya was the "teacher par excellence": he wrote top

books on multiple subjects; his successful students

included John von Neumann.

His work on plane symmetry groups directly inspired Escher's drawings.

Having huge breadth and influence, Pólya has been called

"the most influential mathematician of the 20th century."

__Stefan Banach__

Stefan Banach was a self-taught mathematician

who is most noted as the "Founder of Functional Analysis"

and for his contributions to measure theory.

Among several important theorems bearing his name

are the Uniform Boundedness (Banach-Steinhaus) Theorem,

the Open Mapping (Banach-Schauder) Theorem,

the Contraction Mapping (Banach fixed-point) Theorem,

and the Hahn-Banach Theorem.

Many of these theorems are of practical value to

modern physics; however he also proved

the paradoxical Banach-Tarski Theorem, which demonstrates a

sphere being rearranged intodeuxspheres of the

same original size.

(Banach's proof uses the Axiom of Choice and is sometimes

cited as evidence that that Axiom is false.)

The wide range of Banach's work is indicated by the

Banach-Mazur results in game theory (which also challenge the

axiom of choice).

Banach also made brilliant contributions to probability theory,

set theory, analysis and topology.Banach once said "Mathematics is the most beautiful and most

powerful creation of the human spirit."

__Norbert Wiener__

Norbert Wiener entered college at age 11,

studying various sciences;

he wrote a PhD dissertation at age 17 in philosophy of mathematics where

he was one of the first to show a definition of ordered pair as a set.

(Hausdorff also proposed such a definition; both Wiener's

and Hausdorff's definitions have been superseded by Kuratowski's

(a, b) = a, a, bdespite that it leads

to a singleton when a=b.)

He then did important work in several topics in applied mathematics, including

stochastic processes (beginning with Brownian motion),

potential theory, Fourier analysis,

the Wiener-Hopf decomposition useful for solving

differential and integral equations,

communication theory, cognitive science, and quantum theory.

Many theorems and concepts are named after him, e.g the Wiener Filter

used to reduce the error in noisy signals.

His most important contribution to pure mathematics was his

generalization of Fourier theory into generalized harmonic analysis,

but he is most famous for his writings on feedback

in control systems, for which he coined the new word,cybernetics.

Wiener was first to relate information to thermodynamic entropy,

and anticipated the theory of information attributed to Claude Shannon.

He also designed an early analog computer.

Although they differed dramatically in both personal and mathematical

outlooks, he and John von Neumann were the two key

pioneers (after Turing) in computer science.

Wiener applied his cybernetics to draw conclusions about human

society which, unfortunately, remain largely unheeded.

__Carl Ludwig Siegel__

Carl Siegel became famous when his doctoral dissertation

established a key result in Diophantine approximations.

He continued with contributions

to several branches of analytic and algebraic number theory,

including arithmetic geometry and quadratic forms.

He also did seminal work with Riemann's zeta function,

Dedekind's zeta functions,

transcendental number theory, discontinuous groups,

the 3-body problem in celestial mechanics,

and symplectic geometry.

In complex analysis he developed Siegel modular forms, which have

wide application in math and physics.

He may share credit with Alexander Gelfond for the

solution to Hilbert's 7th Problem.

Siegel admired the "simplicity and honesty" of masters like Gauss, Lagrange and

Hardy and lamented the modern "trend for senseless abstraction."

He and Israel Gelfand were the

first two winners of the Wolf Prize in Mathematics.

Atle Selberg called him a "devastatingly impressive" mathematician who

did things that "seemed impossible."

André Weil declared that Siegel was the greatest mathematician

of the first half of the 20th century.

__Pavel Sergeevich Aleksandrov__

Aleksandrov worked in set theory, metric spaces and

several fields of topology, where he developed techniques of

very broad application.

He pioneered the studies of compact and bicompact spaces,

and homology theory. He laid the groundwork for a key theorem

of metrisation.

His most famous theorem may be his discovery about "perfect subsets"

when he was just 19 years old. Much of his work was done in

collaboration with Pavel Uryson and Heinz Hopf.

Aleksandrov was an important teacher; his students included Lev Pontryagin.

__Emil Artin__

(1898-1962) Austria, Germany, U.S.A.

Artin was an important and prolific researcher in

several fields of algebra, including algebraic number theory,

the theory of rings, field theory,

algebraic topology, Galois theory,

a new method of L-series, and geometric algebra.

Among his most famous theorems were Artin's Reciprocity Law,

key lemmas in Galois theory,

and results in his Theory of Braids.

He also produced two very influential conjectures:

his conjecture about the zeta function

in finite fields developed into the field of arithmetic geometry;

Artin's Conjecture on primitive roots inspired much work

in number theory, and was later generalized to become Weil's Conjectures.

He is credited with solution to Hilbert's 17th Problem

and partial solution to the 9th Problem.

His prize-winning students include John Tate and Serge Lang.

Artin also did work in physical sciences, and was an accomplished musician.

__Paul Adrien Maurice Dirac__

Dirac had a severe father and was bizarrely taciturn

(perhaps autistic), but became one of the greatest mathematical physicists ever.

He developed Fermi-Dirac statistics, applied quantum theory

to field theory, predicted the existence of magnetic monopoles,

and was first to note that some quantum equations

lead to inexplicable infinities.

His most important contribution was to combine relativity and

quantum mechanics by developing, with pure thought, the Dirac Equation.

From this equation, Dirac deduced the existence of anti-electrons,

a prediction considered so bizarre it was ignored —

until anti-electrons were discovered in a cloud chamber four years later.

For this work he was awarded the Nobel Prize in Physics at age 31,

making him one of the youngest Laureates ever.

Dirac's mathematical formulations, including his Equation and the

Dirac-von Neumann axioms, underpin all of modern particle physics.

After his great discovery, Dirac continued to do important work,

some of which underlies modern string theory.

He was also adept at more practical physics; although he declined an invitation

to work on the Manhattan Project, he did contribute a fundamental

result in centrifuge theory to that Project.The Dirac Equation was one of the most important scientific

discoveries of the 20th century and Dirac was certainly a

superb mathematical genius, but I've left Dirac off of the Top 100 since

his work didn't advance "pure" mathematics.

Like many of the other greatest mathematical physicists

(Kepler, Einstein, Weyl), Dirac thought the true equations

of physics must have beauty, writing

"… it is more important to have beauty in one's

equations than to have them fit experiment … (any discrepancy may)

get cleared up with further development of the theory."

__Alfred Tarski__

Alfred Tarski (born Alfred Tajtelbaum)

was one of the greatest and most prolific logicians ever,

but also made advances in set theory, measure theory, topology,

algebra, group theory, computability theory, metamathematics, and geometry.

He was also acclaimed as a teacher.

Although he achieved fame at an early age with the Banach-Tarski

Paradox, his greatest achievements were in formal logic.

He wrote on the definition of truth, developed model theory, and

investigated the completeness questions which also intrigued Gödel.

He proved several important systems

to be incomplete, but also established completeness results for

real arithmetic and geometry.

His most famous result may be

Tarski's Undefinability Theorem, which is related to

Gödel's Incompleteness Theorem but more powerful.

Several other theorems, theories and paradoxes are named after

Tarski including Tarski-Grothendieck Set Theory,

Tarski's Fixed-Point Theorem of lattice theory (from which

the famous Cantor-Bernstein-Schröder Theorem is a simple corollary),

and a new derivation of the Axiom of Choice

(which Lebesgue refused to publish because "an implication between two

false propositions is of no interest").

Tarski was first to enunciate the remarkable fact that the

Generalized Continuum Hypothesis implies the Axiom of Choice,

although proof had to wait for Sierpinski.

Tarski's other notable accomplishments include his cylindrical

algebra, ordinal algebra,

universal algebra, and an elegant and novel axiomatic basis of geometry.

__Jean von Neumann__

John von Neumann (born Neumann Janos Lajos)

was an amazing childhood prodigy who could do

very complicated mental arithmetic and much more at an early age.

One of his teachers burst into tears at their first meeting,

astonished that such a genius existed.

As an adult he was noted for hedonism and reckless driving

but also became one of the most prolific thinkers in history,

making major contributions in many branches of

both pure and applied mathematics.

He was an essential pioneer of both quantum physics and computer science.Von Neumann pioneered the use of models in set theory,

thus improving the axiomatic basis of mathematics.

He proved a generalized spectral theorem sometimes called

the most important result in operator theory.

He developed von Neumann Algebras.

He was first to state and prove the Minimax Theorem and

thus invented game theory; this work also advanced operations research;

and led von Neumann to propose the Doctrine of Mutual Assured

Destruction which was a basis for Cold War strategy.

He developed cellular automata (first invented by Stanislaw Ulam),

famously constructing a self-reproducing automaton.

He worked in mathematical foundations: he formulated

the Axiom of Regularity and invented elegant definitions for the

counting numbers (0 =,n+1 = n ∪ n).

He also worked in analysis, matrix theory,

measure theory, numerical analysis, ergodic theory, group representations,

continuous geometry, statistics and topology.

Von Neumann discovered an ingenious area-conservation paradox

related to the famous Banach-Tarski volume-conservation paradox.

He inspired some of Gödel's famous work

(and independently proved Gödel's Second Theorem).

He is credited with (partial) solution to

Hilbert's 5th Problem using the Haar Theorem;

this also relates to quantum physics.

George Pólya once said

"Johnny was the only student I was ever afraid of.

If in the course of a lecture I stated an unsolved problem,

the chances were he'd come to me as soon as the lecture was over,

with the complete solution in a few scribbles on a slip of paper."

Michael Atiyah has said he calls only three people geniuses:

Wolfgang Mozart, Srinivasa Ramanujan, and Johnny von Neumann.Von Neumann did very important work in fields other than

pure mathematics.

By treating the universe as a very high-dimensional phase space,

he constructed an elegant mathematical basis

(now called von Neumann algebras)

for the principles of quantum physics.

He advanced philosophical questions about time and

logic in modern physics.

He played key roles in the design of conventional, nuclear and thermonuclear

bombs; he also advanced the theory of hydrodynamics.

He applied game theory and Brouwer's Fixed-Point Theorem

to economics, becoming a major figure in that field.

His contributions to computer science are many:

in addition to co-inventing the stored-program computer,

he was first to use pseudo-random number generation,

finite element analysis, the merge-sort algorithm,

a "biased coin" algorithm, and (though Ulam first conceived the

approach) Monte Carlo simulation.

By implementing wide-number software he joined several other

great mathematicians (Archimedes, Apollonius, Liu Hui, Hipparchus,

Madhava, and (by proxy), Ramanujan)

in producing the best approximation toof his time.π

At the time of his death, von Neumann was working on

a theory of the human brain.

__Andrey Nikolaevich Kolmogorov__

Kolmogorov had a powerful intellect and excelled in

many fields.

As a youth he dazzled his teachers by constructing

toys that appeared to be "Perpetual Motion Machines."

At the age of 19, he achieved fame by finding a Fourier series

that diverges almost everywhere, and decided to devote

himself to mathematics.

He is considered the founder of the fields of intuitionistic logic,

algorithmic complexity theory,

and (by applying measure theory) modern probability theory.

He also excelled in topology, set theory, trigonometric series,

and random processes.

He and his student Vladimir Arnold proved the surprising

Superposition Theorem, which not only solved Hilbert's 13th Problem, but

went far beyond it.

He and Arnold also developed the "magnificent" Kolmogorov-Arnold-Moser

(KAM) Theorem,

which quantifies how strong a perturbation must be to upset

a quasiperiodic dynamical system.

Kolmogorov's axioms of probability are considered a partial solution of

Hilbert's 6th Problem.

He made important contributions to the constructivist ideas

of Kronecker and Brouwer.

While Kolmogorov's work in probability theory had direct

applications to physics, Kolmogorov also did work in physics directly,

especially the study of turbulence.

There are dozens of notions named after Kolmogorov,

such as the Kolmogorov Backward Equation,

the Chapman-Kolmogorov equations,

the Borel-Kolmogorov Paradox,

and the intriguing Zero-One Law of "tail events" among random variables.

__Henri Paul Cartan__

Henri Cartan, son of the great Élie Cartan,

is particularly noted for his work in algebraic topology, and

analytic functions; but also worked with sheaves, and many

other areas of mathematics.

He was a key member of the Bourbaki circle.

(That circle was led by Weil,

emphasized rigor, produced important texts, and introduced

terms likein-, sur-,etbi-jection, as well

as theØsymbol.)

Working with Samuel Eilenberg (also a Bourbakian),

Cartan advanced the theory of homological algebra.

He is most noted for his many contributions

to the theory of functions of several complex variables.

Henri Cartan was an important influence on Grothendieck

and others, and an excellent teacher;

his students included Jean-Pierre Serre.

__Kurt Gödel__

Gödel, who had the nickname

Herr Warum("Mr. Why")

as a child, was perhaps the foremost logic theorist

ever, clarifying the relationships between various modes

of logic. He partially resolved both Hilbert's 1st and 2nd Problems,

the latter with a proof so remarkable that it was connected to the

drawings of Escher and music of Bach in the title of a famous book.

He was a close friend of Albert Einstein, and was first to discover

"paradoxical" solutions (e.g. time travel) to Einstein's equations.

About his friend, Einstein later said that he had remained at

Princeton's Institute for Advanced Study merely

"to have the privilege of walking home with Gödel."

(Like a few of the other greatest 20th-century

mathematicians, Gödel was very eccentric.)Two of the major questions confronting mathematics are:

(1) are its axioms consistent (its theorems all being

true statements)?,

and (2) are its axioms complete (its true statements all being theorems)?

Gödel turned his attention to these fundamental questions.

He proved that first-order logic was indeedfullstendig, but that

the more powerful axiom systems needed for arithmetic (constructible

set theory) were necessarilyufullstendig.

He also proved that the Axioms of Choice (AC) and the Generalized Continuum

Hypothesis (GCH) werekonsistentwith set theory, but that set

theory's own consistency could not be proven.

He may have established that the truths of AC and GCH wereuavhengig

of the usual set theory axioms, but the proof was left to Paul Cohen.In Gödel's famous proof of Incompleteness,

he exhibits a true statement (sol) which

cannot be proven, to wit

""sol(this statement itself) cannot be proven.

sisolcould be proven it would be a contradictory

true statement, so consistency dictates that it indeedne peut pas

be proven. But that's whatsolsays, sosolis true!

This sounds like mere word play, but building from ordinary logic

and arithmetic Gödel was able to construct statementsolrigorously.

__André Weil__

Weil made profound contributions to several areas of mathematics,

especially algebraic geometry, which he showed to have deep connections

with number theory.

HisWeil conjectureswere very influential; these and other

works laid the groundwork for some of Grothendieck's work.

Weil proved a special case of the Riemann Hypothesis; he contributed,

at least indirectly, to the recent proof of Fermat's Last Theorem;

he also worked in group theory, general and algebraic topology,

differential geometry, sheaf theory, representation theory, and

theta functions.

He invented several new concepts including vector bundles,

and uniform space.

His work has found applications in particle physics and string theory.

He is considered to be one of the most influential of

modern mathematicians.Weil's biography is interesting. He studied Sanskrit as a child,

loved to travel, taught at a Muslim university in India for

two years (intending to teach French civilization),

wrote as a young man under the famous pseudonym

Nicolas Bourbaki, spent time in prison during World War II

as a Jewish objector, was almost executed as a spy, escaped to

America, and eventually joined Princeton's

Institute for Advanced Studies.

He once wrote:

"Every mathematician worthy of the name has experienced (a)

lucid exaltation in which one thought succeeds another as if miraculously."

__Shiing-Shen Chern__

Shiing-Shen Chern (Chen Xingshen) studied under

Élie Cartan, and became perhaps the greatest

master of differential geometry.

He is especially noted for his work in algebraic geometry, topology

and fiber bundles, developing his Chern characters

(in a paper with "a tremendous number of geometrical jewels"),

developing Chern-Weil theory,

the Chern-Simons invariants,

and especially for his brilliant generalization of

the Gauss-Bonnet Theorem to multiple dimensions.

His work had a major influence in several fields of

modern mathematics as well as gauge theories of physics.

Chern was an important influence in China and a highly

renowned and successful teacher:

one of his students (Yau) won the Fields Medal,

another (Yang) the Nobel Prize in physics.

Chern himself was the first Asian to win the prestigious Wolf Prize.

__Alan Mathison Turing__

Turing developed a new foundation for mathematics

based on computation;

he invented the abstract Turing machine,

designed a "universal" version of such a machine,

proved the famous Halting Theorem (related to

Gödel's Incompleteness Theorem), and

developed the concept of machine intelligence

(including his famousTuring Testproposal).

He also introduced the notions ofdefinable numberet

orakel(important in modern computer science),

and was an early pioneer in the study of neural networks.

For this work he is called the Father

of Computer Science and Artificial Intelligence.

Turing also worked in group theory, numerical analysis,

and complex analysis; he developed an

important theorem about Riemann's zeta function;

he had novel insights in quantum physics.

During World War II he turned his talents to cryptology;

his creative algorithms were considered possibly

"indispensable" to the decryption of German Naval Enigma coding,

which in turn is judged to have certainly shortened the War by at

least two years.

Although his clever code-breaking algorithms were

his most spectacular contributions at Bletchley Park,

he was also a key designer of the Bletchley "Bombe" computer.

After the war he helped design other physical computers,

as well as theoretical designs;

and helped inspire von Neumann's later work.

He (and earlier, von Neumann) wrote about the Quantum Zeno

Effect which is sometimes called the Turing Paradox.

He also studied the mathematics of biology,

especially theTuring Patternsof morphogenesis

which anticipated the discovery of BZ reactions.

Turing's life ended tragically:

charged with immorality and forced to undergo chemical

castration, he apparently took his own life.

With his outstanding depth and breadth, Alan Turing would

qualify for our list in any event, but his decisive contribution to the war

against Hitler gives him unusually strong historic importance.

__Paul Erdös__

(1913-1996) Hungary, U.S.A., Israel, etc.

Erdös was a childhood prodigy who became a

famous (and famously eccentric) mathematician.

He is best known for work in combinatorics (especially Ramsey Theory)

and partition calculus, but made contributions

across a very broad range of mathematics, including

graph theory, analytic number theory, probabilistic methods,

and approximation theory.

He is regarded as the second most prolific mathematician in history,

behind only Euler.

Although he is widely regarded as an important and influential

mathematician, Erdös founded no new field of mathematics:

He was a "problem solver" rather than a "theory developer."

He's left us several still-unproven intriguing conjectures, e.g.

que4/n = 1/x + 1/y + 1/zhas positive-integer solutions

pour certainsn.Erdös liked to speak of "God's Book of Proofs" and discovered new,

more elegant, proofs of several existing theorems,

including the two most famous and important about prime numbers:

Chebyshev's Theorem that there is always a prime

between anynet2net

(though the major contributor was Atle Selberg)

Hadamard's Prime Number Theorem itself.

He also proved many new theorems, such as

the Erdös-Szekeres Theorem about monotone subsequences

with its elegant (if trivial) pigeonhole-principle proof.

__Samuel Eilenberg__

Eilenberg is considered a founder of category theory,

but also worked in algebraic topology, automata theory and other areas.

He coined several new terms includingfunctor,

kategorietnatural isomorphism.

Several other concepts are named after him, e.g. a proof

method called theEilenberg telescope

ouEilenberg-Mazur Swindle.

He worked on cohomology theory, homological algebra, etc.

By using his category theory and axioms of homology, he unified

and revolutionized topology.

Most of his work was done in collaboration with others, e.g.

Henri Cartan; but he also single-authored an important text laying

a mathematical foundation for theories of computation and language.

Sammy Eilenberg was also a noted art collector.

__Israel Moiseevich Gelfand__

Gelfand was a brilliant and important mathematician

of outstanding breadth with a huge number of theorems and discoveries.

He was a key figure of functional analysis and integral geometry;

he pioneered representation theory, important to modern physics;

he also worked in many fields of analysis,

soliton theory, distribution theory, index theory, Banach algebra,

cohomology, etc.

He made advances in physics and biology as well as mathematics.

He won the Order of Lenin three times and several prizes from

Western countries.

Considered one of the two greatest Russian mathematicians of

the 20th century,

the two were compared with

"(arriving in a mountainous country)

Kolmogorov would immediately try to climb the highest mountain;

Gelfand would immediately start to build roads."

In old age Israel Gelfand emigrated to the U.S.A. as a professor,

and won a MacArthur Fellowship.

__Claude Elwood Shannon__

Shannon's initial fame was for a

paper called "possibly the most important master's thesis of the century."

That paper founded digital circuit design theory by proving that universal

computation was achieved with an ensemble of switches and boolean gates.

He also worked with analog computers, theoretical genetics, and sampling

and communication theories.

Early in his career Shannon was fortunate to work with several other

great geniuses including Weyl, Turing, Gödel and even Einstein;

this may have stimulated him toward a broad range of interests and expertise.

He was an important and prolific inventor, discovering signal-flow graphs,

the topological gain formula, etc.; but also inventing the first

wearable computer (to time roulette wheels in Las Vegas casinos),

a chess-playing algorithm, a flame-throwing trumpet,

and whimsical robots (e.g. a "mouse" that navigated a maze).

His hobbies included juggling, unicycling, blackjack card-counting.

His investigations into gambling theory led to new approaches to the stock market.Shannon worked in cryptography during World War II; he was first to

note that a one-time pad allowed unbreakable encryption as long as the pad was

as large as the message; he is also noted for Shannon's maxim that a

code designer should assume the enemy knows the system.

His insights into cryptology eventually led to

information theory, or the mathematical theory of communication, in which

Shannon established the relationships among bits, entropy, power and noise.

It is as the Founder of Information Theory that Shannon has become immortal.

__Atle Selberg__

Selberg may be the greatest analytic number theorist ever.

He also did important work in

Fourier spectral theory,

lattice theory (e.g. introducing and partially

proving the conjecture that "all lattices are arithmetic"),

and the theory of automorphic forms, where he introduced

Selberg's Trace Formula.

He developed a very important result in analysis called

the Selberg Integral.

Other Selberg techniques of general utility include

mollification, sieve theory, and the Rankin-Selberg method.

These have inspired other mathematicians, e.g.

contributing to Deligne's proof of the Weil conjectures.

Selberg is also famous for

ground-breaking work on Riemann's Hypothesis, and

the first "elementary" proof of the Prime Number Theorem.

__John Torrence Tate__

Tate, a student of Emil Artin, is a master of algebraic

number theory, p-adic theory and arithmetic geometry.

Using Fourier analysis and Tate cohomology groups, he

revolutionized the treatments of class field theory and

algebraic K-theory.

In addition to Tate cohomology groups, Tate's key inventions

include rigid analytic geometry, Hodge-Tate theory, Tate-Barsotti groups,

applications of adele ring self-duality, the

Tate module, Tate curve, Tate twists, and much more.

His long and productive career earned the Abel Prize

for his "vast and lasting impact on the theory of numbers

(and) his incisive contributions and illuminating insights …

He has truly left a conspicuous imprint on modern mathematics."

__Jean-Pierre Serre__

Serre did important work with spectral sequences

and algebraic methods,

revolutionizing the study of algebraic topology and algebraic geometry,

especially homotopy groups and sheaves.

Hermann Weyl praised Serre's work strongly, saying it

gave an important new algebraic basis to analysis.

He collaborated with Grothendieck and Pierre Deligne, helped resolve the

Weil conjectures, and contributed indirectly to

the recent proof of Fermat's Last Theorem.

His wide range of research areas also includes

number theory, bundles, fibrations, p-adic modular forms,

Galois representation theory, and more.

Serre has been much honored: he is the youngest ever to win

a Fields Medal; 49 years after his Fields Medal he became

the first recipient of the Abel Prize.

__Alexandre Grothendieck__

Grothendieck has done brilliant work in several areas of mathematics

including number theory, geometry, topology,

and functional analysis,

but especially in the fields of algebraic geometry

and category theory, both of which he revolutionized.

He is especially noted for his invention of the Theory of Schemes,

and other methods to unify different branches of mathematics.

He applied algebraic geometry to number theory;

applied methods of topology to set theory; etc.

Grothendieck is considered a master of abstraction, rigor and presentation.

He has produced many important and deep results in homological algebra,

most notably his etale cohomology.

With these new methods, Grothendieck and his outstanding student Pierre Deligne

were able to prove the Weil Conjectures.

Grothendieck also developed the theory of sheafs, the theory of motives,

generalized the Riemann-Roch Theorem to revolutionize K-theory,

developed Grothendieck categories, crystalline cohomology,

infinity-stacks and more.

The guiding principle behind much of Grothendieck's work has been

Topos Theory, which he invented to harness the methods of topology.

These methods and results have redirected several diverse branches

of modern mathematics including number theory, algebraic topology,

and representation theory.

Among Grothendieck's famous results was his Fundamental Theorem

in the Metric Theory of Tensor Products, which was inspired by

Littlewood's proof of the4/3 Inequality.Grothendieck's radical religious and political philosophies

led him to retire from public life while still in his prime, but

he is widely regarded as the greatest mathematician of the 20th

century, and indeed one of the greatest geniuses ever.

__John Forbes Nash, Jr.__

ils

Riemann Embedding Problemswere important puzzles

of geometry that baffled many of the greatest minds for a century.

Hilbert showed that Lobachevsky's hyperbolic plane

could not be embedded into Euclidean 3-space, but what about into

Euclidean 4-space?

Cartan and Chern were among the great mathematicians who solved

various special cases, but using "methods entirely without precedent"

John Nash demonstrated a general solution.

This was a true highlight of 20th-century mathematics.Nash was a lonely, tormented schizophrenic whose life

was portrayed in the filmBeautiful Mind.

He achieved early fame with his work in game theory; this eventually

led to the Nobel Prize in Economics.

Earlier studies in game theory focused on the simplest

cases (two-person zero-sum, or cooperative), but Nash

demonstrated "Nash equilibria" for n-person or non-zero-sum

non-cooperative games.

Nash also excelled at several other fields of mathematics,

especially topology, algebraic geometry, partial differential equations,

elliptic functions, and the theory of manifolds (including

singularity theory, the concept of real algebraic manifolds

and isotropic embeddings).

He proved theorems of great importance which had defeated all

earlier attempts.

His most famous theorems were the Nash Embedding Theorems,

e.g. that any Riemannian manifold of dimensionkcan be embedded

isometrically into some n-dimensional Euclidean space.

Other important work was in partial differential equations where

he proved that strong regularity constraints apply to

solutions of the equations of heat and fluid flow.

__Lennart Axel Edvard Carleson__

Carleson is a master of complex analysis, especially

harmonic analysis, and dynamical systems; he proved many difficult

and important theorems;

among these are a theorem about

quasiconformal mapping extension, a technique to construct higher

dimensional strange attractors,

and the famous Kakutani Corona Conjecture, whose proof brought

Carleson great fame.

For the Corona proof he introduced Carleson measures, one of several useful

tools he's created for his masterful proofs.

In 1966, four years after proving Kakutani's Conjecture,

he proved the 53-year old Luzin's Conjecture,

a strong statement about Fourier convergence.

This was startling because of a 38-year old conjecture suggested

by Kolmogorovthat Luzin's Conjecture was false.

__Michael Francis (Sir)
Atiyah__

Atiyah's career had extraordinary breadth and depth; han var

sometimes called the greatest English mathematician since Isaac Newton.

He advanced the theory of vector bundles;

this developed into topological K-theory

and the Atiyah-Singer Index Theorem.

This Index Theorem is considered one of the most

far-reaching theorems ever, subsuming famous old results (Descartes'

total angular defect, Euler's topological characteristic),

important 19th-century theorems (Gauss-Bonnet, Riemann-Roch),

and incorporating important work by Weil and especially Shiing-Shen Chern.

It is a key to the study of high-dimension

spaces, differential geometry, and equation solving.

Several other key results are named after Atiyah,

e.g. the Atiyah-Bott Fixed-Point Theorem,

the Atiyah-Segal Completion Theorem,

and the Atiyah-Hirzebruch spectral sequence.

Atiyah's work developed important connections not only between

topology and analysis, but with modern physics; Atiyah himself was

a key figure in the development of string theory;

and was a proponent of the recent idea that octonions may

underlie particle physics.

He also studied the physics of instantons and monopoles.

This work, and Atiyah-inspired work in gauge theory, restored

a close relationship between leading edge research

in mathematics and physics.

His interest in physics, and an old theory of von Neumann,

led him, as a very old man, to explore the fine structure

constant of physics.

In September 2018 he announced that this

recent work would lead to a proof of the Riemann Hypothesis

but other mathematicians are quite skeptical.

Nonetheless, Michael Atiyah is still regarded as one of the

very greatest mathematicians of the 20th century.Atiyah was known as a vivacious genius in person, inspiring many,

e.g. Edward Witten.

Atiyah once said a mathematician must sometimes "freely float in the

atmosphere like a poet and imagine the whole universe of possibilities,

and hope that eventually you come down to Earth somewhere else."

He also said "Beauty is an important criterion in mathematics

… It determines what you regard as important and what is not."

### Mathematicians born after 1930

Many very great mathematicians are alive today,

but this webpage emphasizes *Mathematicians of the Past*,

and mini-bios are provided for only a few recently-born mathematicians.

__John Willard
Milnor__

Milnor founded the field of differential topology

and has made other major advances in topology,

algebraic geometry and dynamical systems.

He discovered Milnor maps (related to fiber bundles);

important theorems in knot theory;

the Duality Theorem for Reidemeister Torsion;

the Milnor Attractors of dynamical systems;

a new elegant proof of Brouwer's "Hairy Ball" Theorem; and much more.

He is especially famous for two counterexamples which each

revolutionized topology.

His "exotic" 7-dimensional hyperspheres

gave the first examples of homeomorphic manifolds that were not also

diffeomorphic, and developed the fields of

differential topology andsurgery theory.

Milnor invented certain high-dimensional polyhedra to disprove the

Hauptvermutung("main conjecture") of geometric topology.

While most famous for his exotic counterexamples,

his revolutionary insights into dynamical systems have important value to

practical applied mathematics.

Although Milnor has been called the "Wizard of Higher Dimensions,"

his work in dynamics began with novel insights into very

low-dimensional systems.As Fields, Presidential and (twice) Putnam Medalist, as well as

winner of the Abel, Wolf anddeuxSteele Prizes; Milnor

can be considered the most "decorated" mathematician of the modern era.

__Roger
Penrose__

Roger Penrose is a thinker of great breadth, who has contributed

to biology and philosophy, as well as to mathematics, general relativity and

cosmology.

Some of his earliest work was done in collaboration with his father Lionel,

a polymath and professor of psychiatry who developed the Penrose Square Root Law of

voting theory.

Together, Roger and his father discovered the 'impossible tri-bar' and

an impossible staircase which inspired work by the artist M.C. Escher.

And, in turn, Escher's drawings may have helped inspire Penrose's

most famous discoveries in recreational mathematics: non-periodic tilings.

He soon found such a tiling with just two tile shapes;

the previous record was six shapes.

(Nine years after that, such tilings were observed in nature as "quasi-crystals.")

Penrose has written several successful popular books on science.As a mathematician, Penrose did important work in algebra: he was first to conceive

of the generalized matrix inverse, and used it for novel solutions in linear

algebra and spectral decomposition.

He did more important work in geometry and topology; for example, he proved

theorems about embedding (or "unknotting") manifolds in Euclidean space.

His best mathematics, e.g. the invention of twistor theory, was inspired

by his pursuit of Einstein's general relativity.Penrose is most noted for his very creative work in cosmology, specifically

in the mathematics of gravitation, space-time, black holes and the Big Bang.

He developed new methods to apply spinors and Riemann tensors to gravitation.

His twistor theory was an effort to relate general relativity

to quantum theory; this work advanced both physics and mathematics.

The top physicist Kip Thorne said "Roger Penrose revolutionized the

mathematical tools that we use to analyse the properties of space-time."

Stephen Hawking was an early convert to Penrose's methods; the mathematical

laws of black holes (and the Big Bang)

are called the Penrose-Hawking Singularity Theorems.

Penrose formulated the Censorship Hypotheses about black holes, e.g. the Riemannian

Penrose Inequality and the Weyl Curvature Hypothesis;

and discovered Penrose-Terrell rotation.Penrose has proposed

conformal cyclic cosmology, que dans

the entropy death of one universe, the scaling of time and distance

become arbitrary and the dying universe becomes the big bang for another.

Recently it is proposed

that evidence for this can be seen in the details of

the cosmic microwave background radiation from the early universe.

Many of his theories are extremely controversial:

He claims that Gödel's

Incompleteness Theorem provides insight into human consciousness.

He has developed a detailed theory that quantum effects (involving

the microtubules in neurons) enhance the capability of biologic brains.

__John Griggs
Thompson__

Thompson is the master of finite groups. He achieved

early fame by proving a long-standing conjecture about Frobenius groups.

He followed up by proving (with Walter Feit) that all nonabelian

finite simple groups are of even order. This result, proved in

a 250-page paper, stunned the world of mathematics; it led to

the classification of all finite groups.

Thompson also made major contributions to coding theory,

and to the inverse Galois problem.

His work with Galois groups has been called "the most important

advance since Hilbert's time."

__Robert Phelan
Langlands__

Langlands started by studying semigroups and partial differential

equations but soon switched his attention to representation theory where he

found deep connections between group theory and

automorphic forms; he then used these connections to make profound

discoveries in number theory.

Langlands' methods, collectively called the Langlands Program,

are now central to all of these fields.

The Langlands Dual Group^{L}G revolutionized representation theory

and led to a large number of conjectures.

One of these conjectures is the Principle of Functoriality, of which a partial

proof allowed Langlands to prove a famous conjecture of Artin, and Wiles

to prove Fermat's Last Theorem.

Langlands and others have applied these methods to prove several

other old conjectures, and to formulate new more powerful conjectures.

He has also worked with Eisenstein series, L-functions, Lie groups,

percolation theory, etc.

He mentored several important mathematicians (including Thomas Hales,

mentioned briefly in Pappus' mini-bio).Langlands once wrote "Certainly the best times were when I was alone with

mathematics, free of ambition and pretense, and indifferent to the world."

He was appointed Hermann Weyl Professor at the Institute for Advanced Study

and now sits in the office once occupied by Albert Einstein.

This seems appropriate since, as the man "who reinvented mathematics,"

his advances have sometimes been compared to Einstein's.

__Vladimir Igorevich
Arnold__

Arnold is most famous for solving Hilbert's 13th Problem;

for the "magnificent" Kolmogorov-Arnold-Moser (KAM) Theorem;

and for "Arnold diffusion," which identifies exceptions to the

stability promised by the KAM Theorem.

In addition to dynamical systems theory, Arnold made contributions

to catastrophe theory, topology, algebraic geometry, symplectic geometry,

differential equations, mathematical physics; and was

the essential founder of modern singularity theory.

__John Horton
Conway__

Conway has done pioneering work in a very broad range

of mathematics including knot theory, number theory, group theory,

lattice theory, combinatorial game theory, geometry, quaternions,

tilings, and cellular automaton theory.

He started his career by proving a case of Waring's Problem,

but achieved fame when he discovered the largest then-known

sporadic group (the symmetry group of the Leech lattice);

this sporadic group is now known to be second in size

only to theMonster Group, with which Conway also worked.

Conway's fertile creativity has produced a

cornucopia of fascinating inventions: markable straight-edge

construction of the regular heptagon (a feat also achieved

by Alhazen, Thabit, Vieta and perhaps Archimedes),

a nowhere-continuous function that has the Intermediate Value property,

the Conway box function,

the rational tangle theorem in knot theory,

the aperiodic pinwheel tiling,

a representation of symmetric polyhedra,

the silly but elegantFractranprogramming language,

his chained-arrow notation for large numbers, and many results and

conjectures in recreational mathematics.

He found the simplest proof for Morley's Trisector Theorem (sometimes

called the best result in simple plane geometry since ancient Greece).

He proved an unusual theorem about quantum physics:

"If experimenters have free will, then so do elementary particles."

His most famous construction is

the computationally complete automaton

known as theGame of Life.

His most important theoretical invention, however,

may be hissurreal numbersincorporating infinitesimals;

he invented them to solve combinatorial games like Go,

but they have pure mathematical significance as the

largest possible ordered field.Conway's great creativity and breadth

certainly make him one of the greatest living mathematicians.

Conway has won the Nemmers Prize in Mathematics,

and was first winner of the Pólya Prize.

__Mikhael Leonidovich
Gromov__

Gromov is considered one of the greatest

geometers ever, but he has a unique "soft" approach to geometry

which leads to applications in other fields: Gromov has contributed to

group theory, partial differential equations,

other areas of analysis and algebra, and even mathematical biology.

He is especially famous for his pseudoholomorphic curves;

they revolutionized the study of symplectic manifolds and are important

in string theory.

By applying his geometric ideas to all areas of mathematics,

Gromov has become one of the most influential living mathematicians.

He has proved a very wide variety of theorems: important results

about groups of polynomial growth, theorems essential to Perelman's proof

of the Poincaré conjecture, the nonsqueezing theorem

of Hamiltonian mechanics, theorems of systolic geometry,

and various inequalities and compactness theorems.

Several concepts are named after him, including Gromov-Hausdorff convergence,

Gromov-Witten invariants, Gromov's random groups, Gromov product, etc.

__Pierre René
Deligne__

(1944-) Belgium, France, U.S.A.

Using new ideas about cohomology, in 1974 Pierre Deligne

stunned the world of mathematics with a spectacular proof of the Weil

conjectures. Proof of these conjectures, which were key to further progress

in algebraic geometry, had eluded the great Alexandre Grothendieck.

With his "unparalleled blend of penetrating insights,

fearless technical mastery and dazzling ingenuity," Deligne made other

important contributions to a broad range of mathematics in addition to

algebraic geometry, including algebraic and analytic number theory,

topology, group theory, the Langlands conjectures,

Grothendieck's theory of motives, and Hodge theory.

Deligne also found a partial solution of Hilbert's 21st Problem.

Several ideas are named after him including

Deligne-Lusztig theory, Deligne-Mumford stacks, Fourier-Deligne transform,

the Langlands-Deligne local constant, Deligne cohomology, and at least

eight distinct conjectures.

__Saharon
Shelah__

Shelah has advanced logic, model theory, set theory,

and especially the theory of cardinal numbers.

His work has led to new methods in diverse fields like group theory,

topology, measure theory, stability theory,

algebraic geometry,

Banach spaces, and combinatorics.

He solved outstanding problems like Morley's Problem; proved

the independence of the Whitehead problem; trouvé

a "Jonsson group" and counterexamples to outstanding conjectures;

improved on Arrow's Voting Theorem; and, most famously,

proved key results about singular cardinals.Shelah is the founder of the theories of proper forcing, classification

of models, and possible cofinalities.

He has authored over 800 papers and several books,

making him one of the most prolific mathematicians in history.

He has been described as a "phenomenal mathematician, … produc(ing)

results at a furious pace."

__William Paul
Thurston__

Thurston revolutionized the study of 3-D manifolds;

it was his work which eventually led Perelman

to a proof of Poincaré's Conjecture.

He developed the key results of foliation theory and, with Gromov,

co-founded geometric group theory.

One of his award citations states "Thurston has fantastic geometric

insight and vision: his ideas have completely revolutionized the study

of topology in 2 and 3 dimensions, and brought about a new and fruitful

interplay between analysis, topology and geometry."

__Edward
Witten__

Witten is the world's greatest living physicist, and one of the top

mathematicians. Not only does Witten apply mathematics to solve problems in

physics, but his broad knowledge of physics, especially quantum field

theory, string theory, supersymmetry, and quantum gravity,

has led him to novel connections and insights in abstract geometry and topology,

as well as physics.

His skill with string theory led him to a novel theory of invariants and

allowed him to improve results in knot theory;

his skill with supersymmetry led him to new results in differential geometry.

He has applied quantum field theory to higher-dimensional spaces and

found new insights there.

He has proven several important new theorems of mathematics and

general relativity but also has had unproven

insights which inspired proofs by others.

His discovery that the five competing models of string theory were all

congruent to a single 'M-theory' (sometimes called a 'Theory of Everything')

revolutionized string theory.

Several theorems or concepts are named after Witten, including

Seiberg-Witten theory, the Weinberg-Witten theorem, the Gromov-Witten invariant,

the Witten index, Witten conjecture, Witten-type Topological quantum field theory, etc.Witten started his college career studying fields like history and linguistics.

When he finally switched to math and physics he learned at breathtaking speed.

His fellows do not compare him to other living mathematical physicists; ils

compare him to Einstein, Weyl, Newton and Ramanujan.

__Terence Chi-Shen
Tao__

Tao was a phenomenal child prodigy who has become one of

the most admired living mathematicians. He has made important contributions

to partial differential equations, combinatorics, harmonic analysis,

number theory, group theory, model theory, nonstandard analysis,

random matrices,

the geometry of 3-manifolds, functional analysis, ergodic theory, etc.

and areas of applied math including quantum mechanics,

general relativity, and image processing.

He has been called the first since David Hilbert to

be expert across the entire spectrum of mathematics.

Among his earliest important discoveries were results about the

multi-dimensional Kakeya needle problem, which led to advances in

Fourier analysis and fractals.

In addition to his numerous research papers he has written many highly

regarded textbooks.

One of his prize citations commends his "sheer technical power,

his other-worldly ingenuity for hitting upon new ideas,

and a startlingly natural point of view."Paul Erdös mentored Tao when he was a ten-year old prodigy,

and the two are frequently compared. They are both prolific

problem solvers across many fields, though have founded no new fields.

As with Erdös, much of Tao's work has been done in collaboration:

for example with Van Vu he proved the circular law of random matrices;

with Ben Green he proved the Dirac-Motzkin conjecture and

solved the "orchard-planting problem."

Especially famous is the Green-Tao Theorem that there are arbitrarily long

arithmetic series among the prime numbers (or indeed among any sufficiently

dense subset of the primes). This confirmed an old

conjecture by Lagrange, and was especially remarkable because the

proof fused methods from number theory, ergodic theory, harmonic analysis,

discrete geometry, and combinatorics.

Tao is also involved in recent efforts to attack the

famous Twin Prime Conjecture.

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This page is copyrighted (©) by James Dow Allen, 1998-2019.

En observant les relations entre les solides de Platon, on peut préciser que l’icosaèdre est l’inverse précis du dodécaèdre. C’est-à-dire, si vous connectez les points centraux des douze pentagones qui composent le composant éthérique, vous aurez créé les 12 coins de l’icosaèdre aqueux. nC’est intrigant car ce que nous avons pu observer jusqu’à présent de l’éther indique qu’il se comporte effectivement comme un fluide. Certes, la mesure et l’observation de l’éther s’est vérifiée assez difficile jusqu’à présent, en raison de son omniprésence. Comment mesurer quelque chose dont on ne peut s’échapper ? Et si nous ne pouvons pas le mesurer, de quelle manière pouvons-nous être sûrs qu’il existe ? nNous avons peu de mal à mesurer les autres composants : la masse cinétique de la terre ; les réactions artificiels rendues solubles par l’eau ; la chaleur rayonnante du feu ; les volts du vent électrique. Celles-ci s’observent relativement facilement, ‘ continuellement ouvertes à notre regard ‘ comme elles le font. Mais l’éther super délicat échappe à une détection facile. ‘ n