geometrical method of Archimedes by inscribed or circumscribed polygons. Thus Vieta, about 1580, computed to ten places, Adrianus Romanus (1561-1615), of Louvain, to 15 places, Ludolph van Ceulen (1540-1610) to 35 places. The latter spent years in this computation, and his performance was considered so extraordinary that the numbers were cut on his tombstone in St. Peter's churchyard at Leyden. The tombstone is lost, but a description of it is extant. After him, the value of is often called "Ludolph's number." In the seventeenth century it was perceived that the computations could be greatly simplified by the use of infinite series. Such a series, viz. tan-1 x = x

X3 205
+

3 5 7

gested by James Gregory in 1671.

+

2

was first sug

Perhaps the easiest are the formulæ used by Machin and Dase. Machin's formula is,

The Englishman, Abraham Sharp, a skilful mechanic and computor, for a time assistant to the astronomer Flamsteed, took the arc in Gregory's formula equal to 30°, and calculated π to 72 places in 1705; next year Machin, professor of astronomy in London, gave to 100 places; the Frenchman, De Lagny, about 1719, gave 127 places; the German, Georg Vesa, in 1793, 140 places; the English, Rutherford, in 1841, 208 places (152 correct); the German, Zacharias Dase, in 1844, 205 places; the German, Th. Clausen, in 1847, 250 places; the English, Rutherford, in 1853, 440 places; William Shanks, in 1873, 707 places.1 It may be remarked that these long computations are of no theoretical or practical value. Infinitely more interesting and useful are Lambert's proof of 1761

1 W. W. R. BALL, Math. Recreations and Problems, pp. 171–173. Ball gives bibliographical references.

that is not rational,' and Lindemann's proof that is not algebraical, i.e. cannot be the root of an algebraic equation.

Infinite series by which may be computed were given also by Hutton and Euler. Leonhard Euler (1707-1783), of Basel, contributed vastly to the progress of higher mathematics, but his influence reached down to elementary subjects. He treated trigonometry as a branch of analysis, introduced (simultaneously with Thomas Simpson in England) the now current abbreviations for trigonometric functions, and simplified formulæ by the simple expedient of designating the angles of a triangle by A, B, C, and the opposite sides by a, b, c. In his old age, after he had become blind, he dictated to his servant his Anleitung zur Algebra, 1770, which, though purely elementary, is meritorious as one of the earliest attempts to put the fundamental processes upon a sound basis. An Introduction to the Elements of Algebra,... selected from the Algebra of Euler, was brought out in 1818 by John Farrar of Harvard College.

A question that became prominent toward the close of the eighteenth century was the graphical representation and interpretation of the imaginary, √-1. As with negative numbers, so with imaginaries, no decided progress was made until a picture of it was presented to the eye. In the time of Newton, Descartes, and Euler, the imaginary was still an algebraic fiction. A geometric picture was given by H. Kühn, a teacher in Danzig, in a publication of 1750-1751. Similar efforts were made by the French, Adrien Quentin Buée and J. F. Français, and more especially by Jean Robert Argand (1768–?), of Geneva, who in 1806 published a remarkable Essai.2 But

1 See the proof in Note IV. of Legendre's Geometrie, where it is extended to π2.

2 Consult Imaginary Quantities. Their Geometrical Interpretation. Translated from the French of M. ARGAND by A. S. HARDY, New York, 1881.

all these writings were little noticed, and it remained for the great Carl Friedrich Gauss (1777-1855), of Göttingen, to break down the last opposition to the imaginary. He introduced it as an independent unit co-ordinate to 1, and a + ib as a "complex number." Notwithstanding the acceptance of imaginaries as "numbers" by all great investigators of the nineteenth century, there are still text-books which represent the obsolete view that √1 is not a number or is not a quantity.

Clear ideas on the fundamental principles of algebra were not secured before the nineteenth century. As late as the latter part of the eighteenth century we find at Cambridge, England, opposition to the use of the negative. The view was held that there exists no distinction between arithmetic and algebra. In fact, such writers as Maclaurin, Saunderson, Thomas Simpson, Hutton, Bonnycastle, Bridge, began their treatises with arithmetical algebra, but gradually and disguisedly introduced negative quantities. Early American writers imitated the English. But in the nineteenth century the first principles of algebra came to be carefully investigated by George Peacock, D. F. Gregory, De Morgan. Of continental writers we may mention Augustin Louis Cauchy (1789-1857), Martin Ohm, and especially Hermann Hankel.'

5

6

3

1 See C. WORDSWORTH, Schola Academica: Some Account of the Studies at English Universities in the Eighteenth Century, 1877, p. 68; Teach. and Hist. of Math. in the U. S., pp. 385–387.

2 See his Algebra, 1830 and 1842, and his "Report on Recent Progress in Analysis," printed in the Reports of the British Association, 1833. 3On the Real Nature of Symbolical Algebra," Trans. Roy. Soc. Edinburgh, Vol. XIV., 1840, p. 280.

4 "On the Foundation of Algebra," Cambridge Phil. Trans., VII., 1841, 1842; VIII., 1844, 1847.

5 Analyse Algébrique, 1821, p. 173 et seq.

6 Versuchs eines vollkommen consequenten Systems der Mathematik, 1822, 2d Ed. 1828.

Die Complexen Zahlen, Leipzig, 1867. This work is very rich in historical notes. Most of the bibliographical references on this subject given here are taken from that work.

A flood of additional light has been thrown on this subject by the epoch-making researches of William Rowan Hamilton, Hermann Grassmann, and Benjamin Peirce, who conceived new algebras with laws differing from the laws of ordinary algebra.1

GEOMETRY AND TRIGONOMETRY

Editions of Euclid. Early Researches

With the close of the fifteenth century and beginning of the sixteenth we enter upon a new era. Great progress was made in arithmetic, algebra, and trigonometry, but less prominent were the advances in geometry. Through the study of Greek manuscripts which, after the fall of Constantinople in 1453, came into possession of Western Europe, improved translations of Euclid were secured. At the beginning of this period, printing was invented; books became cheap and plentiful. The first printed edition of Euclid was published in Venice, 1482. This was the translation from the Arabic by Campanus. Other editions of this appeared, at Ulm in 1486, at Basel in 1491. The first Latin edition, translated from the original Greek, by Bartholomæus Zambertus, appeared at Venice in 1505. In it the translation of Campanus is severely criticised. This led Pacioli, in 1509, to bring out an edition, the tacit aim of which seems to have been to exonerate Campanus. Another Euclid edition appeared in Paris, 1516. The first edition of Euclid printed in Greek was brought out in Basel in 1533, edited by Simon Grynæus. For 170 years this was the only Greek text. In 1703 David Gregory brought out at Oxford all the extant works of Euclid in the original.

1 For an excellent historical sketch on Multiple Algebra, see J. W. GIBBS, in Proceed. Am. Ass. for the Adv. of Science, Vol. XXXV., 1886. 2 CANTOR, II., p. 312.

As a complete edition of Euclid, this stood alo when Heiberg and Menge began the publicat and Latin, of their edition of Euclid's work English translation of the Elements was made the Greek by "H. Billingsley, Citizen of Lo English edition of the Elements and the Data v in 1758 by Robert Simson (1687-1768), profess matics at the University of Glasgow. His te recently the foundation of nearly all school edit fers considerably from the original. Simson cor ber of errors in the Greek copies. All these error to be due to unskilful editors, none to Euclid hims English translation of the Greek text was mad Williamson. The first volume appeared at Oxfo the second volume in 1788. School editions of 1 usually contain the first six books, together with and twelfth.

Returning to the time of the Renaissance, we m

1 In the General Dictionary by BAYLE, London, 1735 Billingsley "made great progress in mathematics, by the his friend, Mr. Whitehead, who, being left destitute upon t of the monasteries in the reign of Henry VIII., was receive ley into his family, and maintained by him in his old age i London." Billingsley was rich and was Lord Mayor of Lo Like other scholars of his day, he confounded our Euclid Megara. The preface to the English edition was written a famous astrologer and mathematician. An interesting a is given in the Dictionary of National Biography. De M that Dee had made the entire translation, but this is denied "Billingsley" of this dictionary. At one time it was beli lingsley translated from an Arabic-Latin version, but G. B. ceeded in proving from a folio - once the property of Billin in the library of Princeton College, and containing the Gr 1533, together with some other editions] that Billingsley tr the Greek, not the Latin. See "Note on the First Englis the Am. Jour. of Mathem., Vol. II., 1879.