Book I

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Euclid wrote other books on both elementary and higher geometry, and on the other mathematical subjects known in his day. The elementary geometrical works include the _Data_ and _On Divisions_ (_of figures_), the first of which survives in Greek and the second in Arabic only; also the _Pseudaria_, now lost, which was a sort of guide to fallacies in geometrical reasoning. The treatises on higher geometry are all lost; they include (1) the _Conics_ in four Books, which covered almost the same ground as the first three Books of Apollonius's _Conics_, although no doubt, for Euclid, the conics were still, as with his predecessors, sections of a right-angled, an obtuse-angled, and an acute-angled cone respectively made by a plane perpendiular to a generator in each case; (2) the _Porisms_ in three Books, the importance and difficulty of which can be inferred from Pappus's account of it and the lemmas which he gives for use with it; (3) the _Surface-Loci_, to which again Pappus furnishes lemmas; one of these implies that Euclid assumed as known the focus-directrix property of the three conics, which is absent from Apollonius's _Conics_.

In applied mathematics Euclid wrote (1) the _Phaenomena_, a work on spherical astronomy in which ὁ ὁριζων {ho horizôn} (without κυκλος {kyklos} or any qualifying words) appears for the first time in the sense of _horizon_; (2) the _Optics_, a kind of elementary treatise on perspective: these two treatises are extant in Greek; (3) a work on the Elements of Music. The _Sectio Canonis_, which has come down under the name of Euclid, can, however, hardly be his in its present form.

In the period between Euclid and Archimedes comes Aristarchus of Samos (about 310-230 B. C.), famous for having anticipated Copernicus. Accepting Heraclides's view that the earth rotates about its own axis, Aristarchus went further and put forward the hypothesis that the sun itself is at rest, and that the earth, as well as Mercury, Venus, and the other planets, revolve in circles about the sun. We have this on the unquestionable authority of Archimedes, who was only some twenty-five years later, and who must have seen the book containing the hypothesis in question. We are told too that Cleanthes the Stoic thought that Aristarchus ought to be indicted on the charge of impiety for setting the Hearth of the Universe in motion.

One work of Aristarchus, _On the sizes and distances of the Sun and Moon_, which is extant in Greek, is highly interesting in itself, though it contains no word of the heliocentric hypothesis. Thoroughly classical in form and style, it lays down certain hypotheses and then deduces therefrom, by rigorous geometry, the sizes and distances of the sun and moon. If the hypotheses had been exact, the results would have been correct too; but Aristarchus in fact assumed a certain angle to be 87° which is really 89° 50', and the angle subtended at the centre of the earth by the diameter of either the sun or the moon to be 2°, whereas we know from Archimedes that Aristarchus himself discovered that the latter angle is only 1/2°. The effect of Aristarchus's geometry is to find arithmetical limits to the values of what are really trigonometrical ratios of certain small angles, namely

1/18 > sin 3° > 1/20, 1/45 > sin 1° > 1/60, 1 > cos 1° > 89/90.

The main results obtained are (1) that the diameter of the sun is between 18 and 20 times the diameter of the moon, (2) that the diameter of the moon is between 2/45ths and 1/30th of the distance of the centre of the moon from our eye, and (3) that the diameter of the sun is between 19/3rds and 43/6ths of the diameter of the earth. The book contains a good deal of arithmetical calculation.

Archimedes was born about 287 B. C. and was killed at the sack of Syracuse by Marcellus's army in 212 B. C. The stories about him are well known, how he said 'Give me a place to stand on, and I will move the earth' (πα βω και κινω ταν γαν {pa bô kai kinô tan gan}); how, having thought of the solution of the problem of the crown when in the bath, he ran home naked shouting ἑυρηκα, ἑυρηκα {heurêka, heurêka}; and how, the capture of Syracuse having found him intent on a figure drawn on the ground, he said to a Roman soldier who came up, 'Stand away, fellow, from my diagram.' Of his work few people know more than that he invented a tubular screw which is still used for pumping water, and that for a long time he foiled the attacks of the Romans on Syracuse by the mechanical devices and engines which he used against them. But he thought meanly of these things, and his real interest was in pure mathematical speculation; he caused to be engraved on his tomb a representation of a cylinder circumscribing a sphere, with the ratio 3/2 which the cylinder bears to the sphere: from which we infer that he regarded this as his greatest discovery.

Archimedes's works are all original, and are perfect models of mathematical exposition; their wide range will be seen from the list of those which survive: _On the Sphere and Cylinder_ I, II, _Measurement of a Circle_, _On Conoids and Spheroids_, _On Spirals_, _On Plane Equilibriums_ I, II, the _Sandreckoner_, _Quadrature of the Parabola_, _On Floating Bodies_ I, II, and lastly the _Method_ (only discovered in 1906). The difficult Cattle-Problem is also attributed to him, and a _Liber Assumptorum_ which has reached us through the Arabic, but which cannot be his in its present form, although some of the propositions in it (notably that about the 'Salinon', salt-cellar, and others about circles inscribed in the αρβηλος {arbêlos}, shoemaker's knife) are quite likely to be of Archimedean origin. Among lost works were the _Catoptrica_, _On Sphere-making_, and investigations into polyhedra, including thirteen semi-regular solids, the discovery of which is attributed by Pappus to Archimedes.

Speaking generally, the geometrical works are directed to the measurement of curvilinear areas and volumes; and Archimedes employs a method which is a development of Eudoxus's method of exhaustion. Eudoxus apparently approached the figure to be measured from below only, i. e. by means of figures successively inscribed to it. Archimedes approaches it from both sides by successively inscribing figures and circumscribing others also, thereby compressing them, as it were, until they coincide as nearly as we please with the figure to be measured. In many cases his procedure is, when the analytical equivalents are set down, seen to amount to real _integration_; this is so with his investigation of the areas of a parabolic segment and a spiral, the surface and volume of a sphere, and the volume of any segments of the conoids and spheroids.

The newly-discovered _Method_ is especially interesting as showing how Archimedes originally obtained his results; this was by a clever mechanical method of (theoretically) _weighing_ infinitesimal elements of the figure to be measured against elements of another figure the area or content of which (as the case may be) is known; it amounts to an _avoidance_ of integration. Archimedes, however, would only admit that the mechanical method is useful for finding results; he did not consider them proved until they were established geometrically.

In the _Measurement of a Circle_, after proving by exhaustion that the area of a circle is equal to a right-angled triangle with the perpendicular sides equal respectively to the radius and the circumference of the circle, Archimedes finds, by sheer calculation, upper and lower limits to the ratio of the circumference of a circle to its diameter (what we call π {p}). This he does by inscribing and circumscribing regular polygons of 96 sides and calculating approximately their respective perimeters. He begins by assuming as known certain approximate values for √3, namely 1351/780 > √3 > 265/153, and his calculations involve approximating to the square roots of several large numbers (up to seven digits). The text only gives the results, but it is evident that the extraction of square roots presented no difficulty, notwithstanding the comparative inconvenience of the alphabetic system of numerals. The result obtained is well known, namely 3-1/7 > π {p} > 3-10/71.

The _Plane Equilibriums_ is the first scientific treatise on the first principles of mechanics, which are established by pure geometry. The most important result established in