Bell - The Development of Mathematics

O. Neugebauer, Acta orientalia , Copenhagen, 17, 1938, 169.

Neugebauer and others in A 6 ; R. C. Archibald, Isis , 71, 1936, 63.

The Rhind mathematical papyrus, A. B. Chace, R. C. Archibald, and others, Oberlin, Ohio, 1927-9.

S. Clarke and R. Engelbach, Ancient Egyptian Masonry, Oxford, 1930.

W. W. Struve in A, Quellen , 1.

Used by Antiphon (c. 430 B.C.), Bryson (5th cent. B.C.), Democritus (460?-370? B.C.).

O. Neugebauer, Vorlesungen , 1934, 203, seems to be more generous.

General reference: A (Tropfke), T. L. Heaths editions of Euclid, Aristarchus, Archimedes, Apollonius, and his other writings, all cited in A (Smith); any edition of Platos dialogues; the bibliography in A 8 .

Nothing of first-rate importance issued from the Greeks perfect numbers.

Claims by Hindu historians, B. Datta and A. N. Singh; His. Hindu Math., Lahore, 1935-8. F. Cajori, Scientific Monthly, 9, 1919, 458. Place-system and zero attributed by some to abacus and other primitive apparatus for arithmetic.

V. F. Hopper, Medieval number mysticism, New York, 1939.

Republic, 546. Timaeus , 53-81, for Platos ripest numerology; criticism by Aristotle, Metaphysics, A, 992a-, 1083b, 987b, 1084.

J. H. Jeans.

Republic, 527: ... this knowledge at which geometry aims is of the eternal, and not of the perishing and transient. Frequently quoted with approval by romanticists ; dismissed as rubbish by mathematicians, who know that mathematics is made by human beings for human needs.

Controversial. Dinostratus, c. 330, is said (Tropfke, 4, 200) to have given the first extant proof by the indirect method. Hippocrates reputedly used the more general technique of equivalent reduction of propositions. Arguments ascribing the method to Plato are vague. It is used in discussing Zenos paradoxes.

Mathematical end, 415 A.D., with death of Hypatia.

Arguments in Heaths Aristarchus .

Plato, Thtaetttur, 147. On mathematical objections to certain historical conjectures, see G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, 1938, 42-3, 180-1; Tropfke, 2, 63-4.

Gibbons Decline and fall of the Roman empire.

Moslem refers to a common religion (Mohametanism), whatever the adherents nationality.

More sympathetic estimates in Smith, A 8, Sarton, A 12.

From what is known of him, the Devil could never have been so stupid at figures as Gerbert, his alleged collaborator, was.

Doubtfully credited with a collection of slightly silly puzzle problems. Even for his own timesthe trite historical clichehe was lower than hundredth-rate mathematically.

A 11, 12, among many.

Bacons rudimentary knowledge of mathematics, as represented in his published writings, scarcely guarantees his frequently quoted testimonialmathematics is the gate and key of the sciences, etc. Curiously, the most flattering testimonials for mathematics have been (and still are) the enthusiastic utterances of men who knew very little about the subject. To dispose here of another myth, Leonardo da Vincis (1452-1591) published jottings on mathematics are trivial, even puerile, and show no mathematical talent whatever.

That the last phrase has no meaning, does not detract from whatever else it may have.

Mysil katolica wobec logiki tasplczsnej (catholic thought in modern logic), Studia gnesiana , 15, Posen, 1937.

Readable account in Gibbons Decline and fall.

Religious designation only; included Arabs, Persians, etc.

Chap. 2.

Almost abandoned.

L. E. Dickson, Hist. theory of numbers , Washington, 2, 1920, 347.

F. Cajori, A 9, 96..

Approvingly quoted from H. Hankel, Geschichte der Mathematik, u.s.w., Leipzig, 1874, by B. Datta, Bull. Calcutta Math. Soc., 19, 1928,87-; S. K. Ganguli, ibid., 151-for criticisms of European critics; approved by Sarton, A 12.

F. Cajori, Hist. math. notations , Chicago, 1928, 1,.84..

A 9, 8, 12.

But not, unfortunately, in textbooks and elementary instruction, where masses of dead material and outmoded ways of thinking obscure the little worth knowing or remembering.

Contrary estimates readily available. As elsewhere, we here ignore textbooks and compilations which, however famous and useful in their day, made no substantial addition to mathematics. Likewise for their authors. Thus L. Pacioli (c. 1445-c. 1509, Tuscan), whose great work [1494] Ramming up ... the general mathematical knowledge of his time is a remarkable compilation with almost no originality; A 8.

C. H. Grffes (1799-1873, Swiss) method (1837) preferred by some.

The libel laws prohibit publication of names.

In method, not in time. Eulers work in algebraic equations is in the same pre-Lagrangian tradition.

But not used. Modern algebra proceeds on lines totally different from those in any of Gauss four proofs (of which two are inadequate).

Philosophiae naturalis principia mathtmatica.

Such as Stirlings for n !.

Tropfke, A 13, 92-100, 104.

J. L. Coolidge, Osiris, 1, 1936, 231-50.

Facsimile reproduction and English translation by D. E. Smith and M. L. Latham, Chicago, 1925.

The so-called plane problem of Pappus inspired Descartes and several of his successors: PL;, PM i ( i = 1, ... , r; j = 1, ... , s ) are the line segments drawn in a fixed direction from P to r + s given straight lines; if

PL 1 PL 2 ... PL, = c . PM 1 PM 2 ... PM s ,

where c is constant, to find the locus of P . Newtons ad quatuor lineas is an elegant solution for r = s = 2.

Appendix to first edition of Opticks: Enumeratio linearum tertii ordinis.

Fermat, Oeuvres, 1, 91-100; 3, 161.

The general French opinion; argument against,

Oeuvrts, 1, 170-3; 2, 354, 457.

Heron of Alexandria knew the special case for reflection from a plane.

Methodus ad disquirendum maximum et minimum, Oeuvres , 1.

If x denotes a number, and 1 the unit length, x n denotes the number x n 1.