Archytas (about 430-360 B. C.) wrote on music and the numerical ratios corresponding to the intervals of the tetrachord. He is said to have been the first to write a treatise on mechanics based on mathematical principles; on the practical side he invented a mechanical dove which would fly. In geometry he gave the first solution of the problem of the two mean proportionals, using a wonderful construction in three dimensions which determined a certain point as the intersection of three surfaces, (1) a certain cone, (2) a half-cylinder, (3) an anchor-ring or tore with inner diameter nil.

Theodorus, Plato’s teacher in mathematics, extended the theory of the irrational by proving incommensurability in certain particular cases other than that of the diagonal of a square in relation to its side, which was already known. He proved that the side of a square containing 3 square feet, or 5 square feet, or any non-square number of square feet up to 17 is incommensurable with one foot, in other words that √3, √5 ... √17 are all incommensurable with 1. Theodorus’s proof was evidently not general; and it was reserved for Theaetetus to comprehend all these irrationals in one definition, and to prove the property generally as it is proved in Eucl. X. 9. Much of the content of the rest of Euclid’s Book X (dealing with compound irrationals), as also of Book XIII on the five regular solids, was due to Theaetetus, who is even said to have discovered two of those solids (the octahedron and icosahedron).

Plato (427-347 B. C.) was probably not an original mathematician, but he ‘caused mathematics in general and geometry in particular to make a great advance by reason of his enthusiasm for them’. He encouraged the members of his school to specialize in mathematics and astronomy; e. g. we are told that in astronomy he set it as a problem to all earnest students to find ‘what are the uniform and ordered movements by the assumption of which the apparent motions of the planets[117] may be accounted for’. In Plato’s own writings are found certain definitions, e. g. that of a straight line as ‘that of which the middle covers the ends’, and some interesting mathematical illustrations, especially that in the second geometrical passage in the Meno (86E-87C). To Plato himself are attributed (1) a formula (n²-1)²+(2n)²=(n²+1)² for finding two square numbers the sum of which is a square number, (2) the invention of the method of analysis, which he is said to have explained to Leodamas of Thasos (mathematical analysis was, however, certainly, in practice, employed long before). The solution, attributed to Plato, of the problem of the two mean proportionals by means of a frame resembling that which a shoemaker uses to measure a foot, can hardly be his.

Eudoxus (408-355 B. C.), an original genius second to none (unless it be Archimedes) in the history of our subject, made two discoveries of supreme importance for the further development of Greek geometry.

(1) As we have seen, the discovery of the incommensurable rendered inadequate the Pythagorean theory of proportion, which applied to commensurable magnitudes only. It would no doubt be possible, in most cases, to replace proofs depending on proportions by others; but this involved great inconvenience, and a slur was cast on geometry generally. The trouble was remedied once for all by Eudoxus’s discovery of the great theory of proportion, applicable to commensurable and incommensurable magnitudes alike, which is expounded in Euclid’s Book V. Well might Barrow say of this theory that ‘there is nothing in the whole body of the elements of a more subtile invention, nothing more solidly established’. The keystone of the structure is the definition of equal ratios (Eucl. V, Def. 5); and twenty-three centuries have not abated a jot from its value, as is plain from the facts that Weierstrass repeats it word for word as his definition of equal[118] numbers, and it corresponds almost to the point of coincidence with the modern treatment of irrationals due to Dedekind.

(2) Eudoxus discovered the method of exhaustion for measuring curvilinear areas and solids, to which, with the extensions given to it by Archimedes, Greek geometry owes its greatest triumphs. Antiphon the Sophist, in connexion with attempts to square the circle, had asserted that, if we inscribe successive regular polygons in a circle, continually doubling the number of sides, we shall sometime arrive at a polygon the sides of which will coincide with the circumference of the circle. Warned by the unanswerable arguments of Zeno against infinitesimals, mathematicians substituted for this the statement that, by continuing the construction, we can inscribe a polygon approaching equality with the circle as nearly as we please. The method of exhaustion used, for the purpose of proof by reductio ad absurdum, the lemma proved in Eucl. X. 1 (to the effect that, if from any magnitude we subtract not less than half, and then from the remainder not less than half, and so on continually, there will sometime be left a magnitude less than any assigned magnitude of the same kind, however small): and this again depends on an assumption which is practically contained in Eucl. V, Def. 4, but is generally known as the Axiom of Archimedes, stating that, if we have two unequal magnitudes, their difference (however small) can, if continually added to itself, be made to exceed any magnitude of the same kind (however great).

The method of exhaustion is seen in operation in Eucl. XII. 1-2, 3-7 Cor., 10, 16-18. Props. 3-7 Cor. and Prop. 10 prove that the volumes of a pyramid and a cone are one-third of the prism and cylinder respectively on the same base and of equal height; and Archimedes expressly says that these facts were first proved by Eudoxus.

In astronomy Eudoxus is famous for the beautiful theory of concentric spheres which he invented to explain the apparent [119] motions of the planets and, particularly, their apparent stationary points and retrogradations. The theory applied also to the sun and moon, for each of which Eudoxus employed three spheres. He represented the motion of each planet as produced by the rotations of four spheres concentric with the earth, one within the other, and connected in the following way. Each of the inner spheres revolves about a diameter the ends of which (poles) are fixed on the next sphere enclosing it. The outermost sphere represents the daily rotation, the second a motion along the zodiac circle; the poles of the third sphere are fixed on the latter circle; the poles of the fourth sphere (carrying the planet fixed on its equator) are so fixed on the third sphere, and the speeds and directions of rotation so arranged, that the planet describes on the second sphere a curve called the hippopede (horse-fetter), or a figure of eight, lying along and longitudinally bisected by the zodiac circle. The whole arrangement is a marvel of geometrical ingenuity.

Heraclides of Pontus (about 388-315 B. C.), a pupil of Plato, made a great step forward in astronomy by his declaration that the earth rotates on its own axis once in 24 hours, and by his discovery that Mercury and Venus revolve about the sun like satellites.

Menaechmus, a pupil of Eudoxus, was the discoverer of the conic sections, two of which, the parabola and the hyperbola, he used for solving the problem of the two mean proportionals. If a:x=x:y=y:b, then x²=ay, y²=bx and xy=ab. These equations represent, in Cartesian co-ordinates, and with rectangular axes, the conics by the intersection of which two and two Menaechmus solved the problem; in the case of the rectangular hyperbola it was the asymptote-property which he used.

We pass to Euclid’s times. A little older than Euclid, Autolycus of Pitane wrote two books, On the Moving Sphere, a work on Sphaeric for use in astronomy, and On Risings and [120] Settings. The former work is the earliest Greek textbook which has reached us intact. It was before Euclid when he wrote his Phaenomena, and there are many points of contact between the two books.

Euclid flourished about 300 B. C. or a little earlier. His great work, the Elements in thirteen Books, is too well known to need description. No work presumably, except the Bible, has had such a reign; and future generations will come back to it again and again as they tire of the variegated substitutes for it and the confusion resulting from their bewildering multiplicity. After what has been said above of the growth of the Elements, we can appreciate the remark of Proclus about Euclid, ‘who put together the Elements, collecting many of Eudoxus’s theorems, perfecting many of Theaetetus’s and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors’. Though a large portion of the subject-matter had been investigated by those predecessors, everything goes to show that the whole arrangement was Euclid’s own; it is certain that he made great changes in the order of propositions and in the proofs, and that his innovations began at the very beginning of Book I.

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 [121] 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 (without κυκλος 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 [122] 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 ½°. 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; how, having thought of the solution of the problem of the crown when in the bath, he ran home naked shouting ἑυρηκα, ἑυρηκα; 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 [123] 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, 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.

[124] 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). 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 > 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 Book I is the principle of the lever. This was known to Plato and Aristotle, but they had no real proof. The Aristotelian Mechanics merely ‘refers’ the lever ‘to the circle’, asserting that the force which acts at the greater distance from the fulcrum moves the system more easily because it describes a greater circle. Archimedes also finds the centre of gravity[125] of a parallelogram, a triangle, a trapezium and finally (in Book II) of a parabolic segment and of a portion of it cut off by a straight line parallel to the base.

The Sandreckoner is remarkable for the development in it of a system for expressing very large numbers by orders and periods based on powers of myriad-myriads (10,000²). It also contains the important reference to the heliocentric theory of the universe put forward by Aristarchus of Samos in a book of ‘hypotheses’, as well as historical details of previous attempts to measure the size of the earth and to give the sizes and distances of the sun and moon.

Lastly, Archimedes invented the whole science of hydrostatics. Beginning the treatise On Floating Bodies with an assumption about uniform pressure in a fluid, he first proves that the surface of a fluid at rest is a sphere with its centre at the centre of the earth. Other propositions show that, if a solid floats in a fluid, the weight of the solid is equal to that of the fluid displaced, and, if a solid heavier than a fluid is weighed in it, it will be lighter than its true weight by the weight of the fluid displaced. Then, after a second assumption that bodies which are forced upwards in a fluid are forced upwards along the perpendiculars to the surface which pass through their centres of gravity, Archimedes deals with the position of rest and stability of a segment of a sphere floating in a fluid with its base entirely above or entirely below the surface. Book II is an extraordinary tour de force, investigating fully all the positions of rest and stability of a right segment of a paraboloid floating in a fluid according (1) to the relation between the axis of the solid and the parameter of the generating parabola, and (2) to the specific gravity of the solid in relation to the fluid; the term ‘specific gravity’ is not used, but the idea is fully expressed in other words.

Almost contemporary with Archimedes was Eratosthenes of Cyrene, to whom Archimedes dedicated the Method; the [126] preface to this work shows that Archimedes thought highly of his mathematical ability. He was indeed recognized by his contemporaries as a man of great distinction in all branches, though the names Beta and Pentathlos[4] applied to him indicate that he just fell below the first rank in each subject. Ptolemy Euergetes appointed him to be tutor to his son (Philopator), and he became librarian at Alexandria; he recognized his obligation to Ptolemy by erecting a column with a graceful epigram. In this epigram he referred to the earlier solutions of the problem of duplicating the cube or finding the two mean proportionals, and advocated his own in preference, because it would give any number of means; on the column was fixed a bronze representation of his appliance, a frame with right-angled triangles (or rectangles) movable along two parallel grooves and over one another, together with a condensed proof. The Platonicus of Eratosthenes evidently dealt with the fundamental notions of mathematics in connexion with Plato’s philosophy, and seems to have begun with the story of the origin of the duplication problem.

The most famous achievement of Eratosthenes was his measurement of the earth. Archimedes quotes an earlier measurement which made the circumference of the earth 300,000 stades. Eratosthenes improved upon this. He observed that at the summer solstice at Syene, at noon, the sun cast no shadow, while at the same moment the upright gnomon at Alexandria cast a shadow corresponding to an angle between the gnomon and the sun’s rays of 1/50th of four right angles. The distance between Syene and Alexandria being known to be 5,000 stades, this gave for the circumference of the earth 250,000 stades, which Eratosthenes seems later, for some reason, to have changed to 252,000 stades. On the [127] most probable assumption as to the length of the stade used, the 252,000 stades give about 7,850 miles, only 50 miles less than the true polar diameter.

In the work On the Measurement of the Earth Eratosthenes is said to have discussed other astronomical matters, the distance of the tropic and polar circles, the sizes and distances of the sun and moon, total and partial eclipses, &c. Besides other works on astronomy and chronology, Eratosthenes wrote a Geographica in three books, in which he first gave a history of geography up to date and then passed on to mathematical geography, the spherical shape of the earth, &c., &c.

Apollonius of Perga was with justice called by his contemporaries the ‘Great Geometer’, on the strength of his great treatise, the Conics. He is mentioned as a famous astronomer of the reign of Ptolemy Euergetes (247-222 B. C.); and he dedicated the fourth and later Books of the Conics to King Attalus I of Pergamum (241-197 B. C.).

The Conics, a colossal work, originally in eight Books, survives as to the first four Books in Greek and as to three more in Arabic, the eighth being lost. From Apollonius’s prefaces we can judge of the relation of his work to Euclid’s Conics, the content of which answered to the first three Books of Apollonius. Although Euclid knew that an ellipse could be otherwise produced, e. g. as an oblique section of a right cylinder, there is no doubt that he produced all three conics from right cones like his predecessors. Apollonius, however, obtains them in the most general way by cutting any oblique cone, and his original axes of reference, a diameter and the tangent at its extremity, are in general oblique; the fundamental properties are found with reference to these axes by ‘application of areas’, the three varieties of which, application (παραβολη), application with an excess (ὑπερβολη) and application with a deficiency (ελλειψις), give the properties of the three curves respectively and account for the names[128] parabola, hyperbola, and ellipse, by which Apollonius called them for the first time. The principal axes only appear, as a particular case, after it has been shown that the curves have a like property when referred to any other diameter and the tangent at its extremity, instead of those arising out of the original construction. The first four Books constitute what Apollonius calls an elementary introduction; the remaining Books are specialized investigations, the most important being Book V (on normals) and Book VII (mainly on conjugate diameters). Normals are treated, not in connexion with tangents, but as minimum or maximum straight lines drawn to the curves from different points or classes of points. Apollonius discusses such questions as the number of normals that can be drawn from one point (according to its position) and the construction of all such normals. Certain propositions of great difficulty enable us to deduce quite easily the Cartesian equations to the evolutes of the three conics.

Several other works of Apollonius are described by Pappus as forming part of the ‘Treasury of Analysis’. All are lost except the Sectio Rationis in two Books, which survives in Arabic and was published in a Latin translation by Halley in 1706. It deals with all possible cases of the general problem ‘given two straight lines either parallel or intersecting, and a fixed point on each, to draw through any given point a straight line which shall cut off intercepts from the two lines (measured from the fixed points) bearing a given ratio to one another’. The lost treatise Sectio Spatii dealt similarly with the like problem in which the intercepts cut off have to contain a given rectangle.

The other treatises included in Pappus’s account are (1) On Determinate Section; (2) Contacts or Tangencies, Book II of which is entirely devoted to the problem of drawing a circle to touch three given circles (Apollonius’s solution can, with the aid of Pappus’s auxiliary propositions, be satisfactorily [129] restored); (3) Plane Loci, i. e. loci which are straight lines or circles; (4) Νευσεις, Inclinationes (the general problem called a νευσις being to insert between two lines, straight or curved, a straight line of given length verging to a given point, i. e. so that, if produced, it passes through the point, Apollonius restricted himself to cases which could be solved by ‘plane’ methods, i. e. by the straight line and circle only).

Apollonius is also said to have written (5) a Comparison of the dodecahedron with the icosahedron (inscribed in the same sphere), in which he proved that their surfaces are in the same ratio as their volumes; (6) On the cochlias or cylindrical helix; (7) a ‘General Treatise’, which apparently dealt with the fundamental assumptions, &c., of elementary geometry; (8) a work on unordered irrationals, i. e. irrationals of more complicated form than those of Eucl. Book X; (9) On the burning-mirror, dealing with spherical mirrors and probably with mirrors of parabolic section also; (10) ωκυτοκιον (‘quick delivery’). In the last-named work Apollonius found an approximation to π closer than that in Archimedes’s Measurement of a Circle; and possibly the book also contained Apollonius’s exposition of his notation for large numbers according to ‘tetrads’ (successive powers of the myriad).

In astronomy Apollonius is said to have made special researches regarding the moon, and to have been called ε (Epsilon) because the form of that letter is associated with the moon. He was also a master of the theory of epicycles and eccentrics.

With Archimedes and Apollonius Greek geometry reached its culminating point; indeed, without some more elastic notation and machinery such as algebra provides, geometry was practically at the end of its resources. For some time, however, there were capable geometers who kept up the tradition, filling in details, devising alternative solutions of problems, or discovering new curves for use or investigation.

[130] Nicomedes, probably intermediate in date between Eratosthenes and Apollonius, was the inventor of the conchoid or cochloid, of which, according to Pappus, there were three varieties. Diocles (about the end of the second century B. C.) is known as the discoverer of the cissoid which was used for duplicating the cube. He also wrote a book περι πυρειων, On burning-mirrors, which probably discussed, among other forms of mirror, surfaces of parabolic or elliptic section, and used the focal properties of the two conics; it was in this work that Diocles gave an independent and clever solution (by means of an ellipse and a rectangular hyperbola) of Archimedes’s problem of cutting a sphere into two segments in a given ratio. Dionysodorus gave a solution by means of conics of the auxiliary cubic equation to which Archimedes reduced this problem; he also found the solid content of a tore or anchor-ring.

Perseus is known as the discoverer and investigator of the spiric sections, i. e. certain sections of the σπειρα, one variety of which is the tore. The spire is generated by the revolution of a circle about a straight line in its plane, which straight line may either be external to the circle (in which case the figure produced is the tore), or may cut or touch the circle.

Zenodorus was the author of a treatise on Isometric figures, the problem in which was to compare the content of different figures, plane or solid, having equal contours or surfaces respectively.

Hypsicles (second half of second century B. C.) wrote what became known as ‘Book XIV’ of the Elements containing supplementary propositions on the regular solids (partly drawn from Aristaeus and Apollonius); he seems also to have written on polygonal numbers. A mediocre astronomical work (Αναφορικος) attributed to him is the first Greek book in which we find the division of the zodiac circle into 360 parts or degrees.

Posidonius the Stoic (about 135-51 B. C.) wrote on geography [131] and astronomy under the titles On the Ocean and περι μετεωρων. He made a new but faulty calculation of the circumference of the earth (240,000 stades). Per contra, in a separate tract on the size of the sun (in refutation of the Epicurean view that it is as big as it looks), he made assumptions (partly guesswork) which give for the diameter of the sun a figure of 3,000,000 stades (39-1/4 times the diameter of the earth), a result much nearer the truth than those obtained by Aristarchus, Hipparchus, and Ptolemy. In elementary geometry Posidonius gave certain definitions (notably of parallels, based on the idea of equidistance).

Geminus of Rhodes, a pupil of Posidonius, wrote (about 70 B. C.) an encyclopaedic work on the classification and content of mathematics, including the history of each subject, from which Proclus and others have preserved notable extracts. An-Nairīzī (an Arabian commentator on Euclid) reproduces an attempt by one ‘Aganis’, who appears to be Geminus, to prove the parallel-postulate.

But from this time onwards the study of higher geometry (except sphaeric) seems to have languished, until that admirable mathematician, Pappus, arose (towards the end of the third century A. D.) to revive interest in the subject. From the way in which, in his great Collection, Pappus thinks it necessary to describe in detail the contents of the classical works belonging to the ‘Treasury of Analysis’ we gather that by his time many of them had been lost or forgotten, and that he aimed at nothing less than re-establishing geometry at its former level. No one could have been better qualified for the task. Presumably such interest as Pappus was able to arouse soon flickered out; but his Collection remains, after the original works of the great mathematicians, the most comprehensive and valuable of all our sources, being a handbook or guide to Greek geometry and covering practically the whole field. Among the original things in Pappus’s Collection is an enunciation[132] which amounts to an anticipation of what is known as Guldin’s Theorem.

It remains to speak of three subjects, trigonometry (represented by Hipparchus, Menelaus, and Ptolemy), mensuration (in Heron of Alexandria), and algebra (Diophantus).

Although, in a sense, the beginnings of trigonometry go back to Archimedes (Measurement of a Circle), Hipparchus was the first person who can be proved to have used trigonometry systematically. Hipparchus, the greatest astronomer of antiquity, whose observations were made between 161 and 126 B. C., discovered the precession of the equinoxes, calculated the mean lunar month at 29 days, 12 hours, 44 minutes, 2½ seconds (which differs by less than a second from the present accepted figure!), made more correct estimates of the sizes and distances of the sun and moon, introduced great improvements in the instruments used for observations, and compiled a catalogue of some 850 stars; he seems to have been the first to state the position of these stars in terms of latitude and longitude (in relation to the ecliptic). He wrote a treatise in twelve Books on Chords in a Circle, equivalent to a table of trigonometrical sines. For calculating arcs in astronomy from other arcs given by means of tables he used propositions in spherical trigonometry.

The Sphaerica of Theodosius of Bithynia (written, say, 20 B. C.) contains no trigonometry. It is otherwise with the Sphaerica of Menelaus (fl. A. D. 100) extant in Arabic; Book I of this work contains propositions about spherical triangles corresponding to the main propositions of Euclid about plane triangles (e.g. congruence theorems and the proposition that in a spherical triangle the three angles are together greater than two right angles), while Book III contains genuine spherical trigonometry, consisting of ‘Menelaus’s Theorem’ with reference to the sphere and deductions therefrom.

Ptolemy’s great work, the Syntaxis, written about A. D. 150[133] and originally called Μαθηματικη συνταξις, came to be known as Μεγαλη συνταξις; the Arabs made up from the superlative μεγιστος the word al-Majisti which became Almagest.

Book I, containing the necessary preliminaries to the study of the Ptolemaic system, gives a Table of Chords in a circle subtended by angles at the centre of ½° increasing by half-degrees to 180°. The circle is divided into 360 μοιραι, parts or degrees, and the diameter into 120 parts (τμηματα); the chords are given in terms of the latter with sexagesimal fractions (e. g. the chord subtended by an angle of 120° is 103^p 53′ 23″). The Table of Chords is equivalent to a table of the sines of the halves of the angles in the table, for, if (crd. 2 α) represents the chord subtended by an angle of 2 α (crd. 2 α)/120 = sin α. Ptolemy first gives the minimum number of geometrical propositions required for the calculation of the chords. The first of these finds (crd. 36°) and (crd. 72°) from the geometry of the inscribed pentagon and decagon; the second (‘Ptolemy’s Theorem’ about a quadrilateral in a circle) is equivalent to the formula for sin (θ-φ), the third to that for sin ½ θ. From (crd. 72°) and (crd. 60°) Ptolemy, by using these propositions successively, deduces (crd. 1½°) and (crd. ¾°), from which he obtains (crd. 1°) by a clever interpolation. To complete the table he only needs his fourth proposition, which is equivalent to the formula for cos (θ+φ).

Ptolemy wrote other minor astronomical works, most of which survive in Greek or Arabic, an Optics in five Books (four Books almost complete were translated into Latin in the twelfth century), and an attempted proof of the parallel-postulate which is reproduced by Proclus.

Heron of Alexandria (date uncertain; he may have lived as late as the third century A. D.) was an almost encyclopaedic writer on mathematical and physical subjects. He aimed at practical utility rather than theoretical completeness; hence, apart from the interesting collection of Definitions which has[134] come down under his name, and his commentary on Euclid which is represented only by extracts in Proclus and an-Nairīzī, his geometry is mostly mensuration in the shape of numerical examples worked out. As these could be indefinitely multiplied, there was a temptation to add to them and to use Heron’s name. However much of the separate works edited by Hultsch (the Geometrica, Geodaesia, Stereometrica, Mensurae, Liber geëponicus) is genuine, we must now regard as more authoritative the genuine Metrica discovered at Constantinople in 1896 and edited by H. Schöne in 1903 (Teubner). Book I on the measurement of areas is specially interesting for (1) its statement of the formula used by Heron for finding approximations to surds, (2) the elegant geometrical proof of the formula for the area of a triangle Δ = √{s (s-a) (s-b) (s-c)}, a formula now known to be due to Archimedes, (3) an allusion to limits to the value of π found by Archimedes and more exact than the 3-1/7 and 3-10/71 obtained in the Measurement of a Circle.

Book I of the Metrica calculates the areas of triangles, quadrilaterals, the regular polygons up to the dodecagon (the areas even of the heptagon, enneagon, and hendecagon are approximately evaluated), the circle and a segment of it, the ellipse, a parabolic segment, and the surfaces of a cylinder, a right cone, a sphere and a segment thereof. Book II deals with the measurement of solids, the cylinder, prisms, pyramids and cones and frusta thereof, the sphere and a segment of it, the anchor-ring or tore, the five regular solids, and finally the two special solids of Archimedes’s Method; full use is made of all Archimedes’s results. Book III is on the division of figures. The plane portion is much on the lines of Euclid’s Divisions (of figures). The solids divided in given ratios are the sphere, the pyramid, the cone and a frustum thereof. Incidentally Heron shows how he obtained an approximation to the cube root of a non-cube number (100). Quadratic equations are solved by Heron by a regular rule not unlike our[135] method, and the Geometrica contains two interesting indeterminate problems.

Heron also wrote Pneumatica (where the reader will find such things as siphons, Heron’s Fountain, penny-in-the-slot machines, a fire-engine, a water-organ, and many arrangements employing the force of steam), Automaton-making, Belopoeïca (on engines of war), Catoptrica, and Mechanics. The Mechanics has been edited from the Arabic; it is (except for considerable fragments) lost in Greek. It deals with the puzzle of ‘Aristotle’s Wheel’, the parallelogram of velocities, definitions of, and problems on, the centre of gravity, the distribution of weights between several supports, the five mechanical powers, mechanics in daily life (queries and answers). Pappus covers much the same ground in Book VIII of his Collection.

We come, lastly, to Algebra. Problems involving simple equations are found in the Papyrus Rhind, in the Epanthema of Thymaridas already referred to, and in the arithmetical epigrams in the Greek Anthology (Plato alludes to this class of problem in the Laws, 819 B, C); the Anthology even includes two cases of indeterminate equations of the first degree. The Pythagoreans gave general solutions in rational numbers of the equations x²+y²=z² and 2x²-y²=±1, which are indeterminate equations of the second degree.

The first to make systematic use of symbols in algebraical work was Diophantus of Alexandria (fl. about A. D. 250). He used (1) a sign for the unknown quantity, which he calls αριθμος, and compendia for its powers up to the sixth; (2) a sign () with the effect of our minus. The latter sign probably represents ΛΙ, an abbreviation for the root of the word λειπειν (to be wanting); the sign for αριθμος () is most likely an abbreviation for the letters αρ; the compendia for the powers of the unknown are Δ^Υ for δυναμις, the square, Κ^Υ for κυβος, the cube, and so on. Diophantus shows that he solved quadratic equations by rule, like Heron. His Arithmetica, of[136] which six books only (out of thirteen) survive, contains a certain number of problems leading to simple equations, but is mostly devoted to indeterminate or semi-determinate analysis, mainly of the second degree. The collection is extraordinarily varied, and the devices resorted to are highly ingenious. The problems solved are such as the following (fractional as well as integral solutions being admitted): ‘Given a number, to find three others such that the sum of the three, or of any pair of them, together with the given number is a square’, ‘To find four numbers such that the square of the sum plus or minus any one of the numbers is a square’, ‘To find three numbers such that the product of any two plus or minus the sum of the three is a square’. Diophantus assumes as known certain theorems about numbers which are the sums of two and three squares respectively, and other propositions in the Theory of Numbers. He also wrote a book On Polygonal Numbers of which only a fragment survives.

With Pappus and Diophantus the list of original writers on mathematics comes to an end. After them came the commentators whose names only can be mentioned here. Theon of Alexandria, the editor of Euclid, lived towards the end of the fourth century A. D. To the fifth and sixth centuries belong Proclus, Simplicius, and Eutocius, to whom we can never be grateful enough for the precious fragments which they have preserved from works now lost, and particularly the History of Geometry and the History of Astronomy by Aristotle’s pupil Eudemus.

Such is the story of Greek mathematical science. If anything could enhance the marvel of it, it would be the consideration of the shortness of the time (about 350 years) within which the Greeks, starting from the very beginning, brought geometry to the point of performing operations equivalent to the integral calculus and, in the realm of astronomy, actually anticipated Copernicus.

T. L. Heath.

---

---

FOOTNOTES: