The method ofreductio ad absurdumis a variety of analysis.Starting from a hypothesis, namely the contradictory of what we desire to prove, we use the same process of analysis, carrying it back until we arrive at something admittedly false or absurd. Aristotle describes this method in various ways asreductio ad absurdum, proofper impossibile, or proof leading to the impossible. But here again, though the term was new, the method was not. The paradoxes of Zeno are classical instances.
Lastly, the Greeks established the form of exposition which still governs geometrical work, simply because it is dictated by strict logic. It is seen in Euclid’s propositions, with their separate formal divisions, to which specific names were afterwards assigned, (1) theenunciation(προτασις), (2) thesetting-out(εκθεσις), (3) theδιορισμος, being a re-statement of what we are required to do or prove, not in general terms (as in theenunciation), but with reference to the particular data contained in thesetting-out, (4) theconstruction(κατασκευη), (5) theproof(αποδειξις), (6) theconclusion(συμπερασμα). In the case of a problem it often happens that a solution is not possible unless the particular data are such as to satisfy certain conditions; in this case there is yet another constituent part in the proposition, namely the statement of the conditions or limits of possibility, which was called by the same nameδιορισμος, definition or delimitation, as that applied to the third constituent part of a theorem.
We have so far endeavoured to indicate generally the finality and the abiding value of the work done by the creators of mathematical science. It remains to summarize, as briefly as possible, the history of Greek mathematics according to periods and subjects.
The Greeks of course took what they could in the shape of elementary facts in geometry and astronomy from the Egyptians and Babylonians. But some of the essential characteristics of the Greek genius assert themselves even intheir borrowings from these or other sources. Here, as everywhere else, we see their directness and concentration; they always knew what they wanted, and they had an unerring instinct for taking only what was worth having and rejecting the rest. This is illustrated by the story of Pythagoras’s travels. He consorted with priests and prophets and was initiated into the religious rites practised in different places, not out of religious enthusiasm ‘as you might think’ (says our informant), but in order that he might not overlook any fragment of knowledge worth acquiring that might lie hidden in the mysteries of divine worship.
This story also illustrates an important advantage which the Greeks had over the Egyptians and Babylonians. In those countries science, such as it was, was the monopoly of the priests; and, where this is the case, the first steps in science are apt to prove the last also, because the scientific results attained tend to become involved in religious prescriptions and routine observances, and so to end in a collection of lifeless formulae. Fortunately for the Greeks, they had no organized priesthood; untrammelled by prescription, traditional dogmas or superstition, they could give their reasoning faculties free play. Thus they were able to create science as a living thing susceptible of development without limit.
Greek geometry, as also Greek astronomy, begins with Thales (about 624-547B. C.), who travelled in Egypt and is said to have brought geometry from thence. Such geometry as there was in Egypt arose out of practical needs. Revenue was raised by the taxation of landed property, and its assessment depended on the accurate fixing of the boundaries of the various holdings. When these were removed by the periodical flooding due to the rising of the Nile, it was necessary to replace them, or to determine the taxable area independently of them, by an art of land-surveying. We conclude from the Papyrus Rhind (say 1700B. C.) and other documents that Egyptian geometryconsisted mainly of practical rules for measuring, with more or less accuracy, (1) such areas as squares, triangles, trapezia, and circles, (2) the solid content of measures of corn, &c., of different shapes. The Egyptians also constructed pyramids of a certain slope by means of arithmetical calculations based on a certain ratio,se-qeṭ, namely the ratio of half the side of the base to the height, which is in fact equivalent to the co-tangent of the angle of slope. The use of this ratio implies the notion of similarity of figures, especially triangles. The Egyptians knew, too, that a triangle with its sides in the ratio of the numbers 3, 4, 5 is right-angled, and used the fact as a means of drawing right angles. But there is no sign that they knew the general property of a right-angled triangle (= Eucl. I. 47), of which this is a particular case, or that they proved any general theorem in geometry.
No doubt Thales, when he was in Egypt, would see diagrams drawn to illustrate the rules for the measurement of circles and other plane figures, and these diagrams would suggest to him certain similarities and congruences which would set him thinking whether there were not some elementary general principles underlying the construction and relations of different figures and parts of figures. This would be in accord with the Greek instinct for generalization and their wish to be able to account for everything on rational principles.
The following theorems are attributed to Thales: (1) that a circle is bisected by any diameter (Eucl. I, Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I. 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I. 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I. 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle, which must mean that he was the first to discover that the angle in a semicircle is a right angle (cf. Eucl. III. 31).
Elementary as these things are, they represent a new departure of a momentous kind, being the first steps towards atheoryof geometry. On this point we cannot do better than quote some remarks from Kant’s preface to the second edition of hisKritik der reinen Vernunft.
‘Mathematics has, from the earliest times to which the history of human reason goes back, (that is to say) with that wonderful people the Greeks, travelled the safe road of ascience. But it must not be supposed that it was as easy for mathematics as it was for logic, where reason is concerned with itself alone, to find, or rather to build for itself, that royal road. I believe on the contrary that with mathematics it remained for long a case of groping about—the Egyptians in particular were still at that stage—and that this transformation must be ascribed to arevolutionbrought about by the happy inspiration of one man in trying an experiment, from which point onward the road that must be taken could no longer be missed, and the safe way of a science was struck and traced out for all time and to distances illimitable.... A light broke on the first man who demonstrated the property of the isosceles triangle (whether his name was Thales or what you will)....’
Thales also solved two problems of a practical kind: (1) he showed how to measure the distance of a ship at sea, and (2) he found the heights of pyramids by means of the shadows thrown on the ground by the pyramid and by a stick of known length at the same moment; one account says that he chose the time when the lengths of the stick and of its shadow were equal, but in either case he argued by similarity of triangles.
In astronomy Thales predicted a solar eclipse which was probably that of the 28th May 585B. C.Now the Babylonians, as the result of observations continued through centuries, had discovered the period of 223 lunations after which eclipsesrecur. It is most likely therefore that Thales had heard of this period, and that his prediction was based upon it. He is further said to have used the Little Bear for finding the pole, to have discovered the inequality of the four astronomical seasons, and to have written worksOn the EquinoxandOn the Solstice.
After Thales come the Pythagoreans. Of the Pythagoreans Aristotle says that they applied themselves to the study of mathematics and were the first to advance that science, going so far as to find in the principles of mathematics the principles of all existing things. Of Pythagoras himself we are told that he attached supreme importance to the study of arithmetic, advancing it and taking it out of the region of practical utility, and again that he transformed the study of geometry into a liberal education, examining the principles of the science from the beginning.
The very wordμαθηματα, which originally meant ‘subjects of instruction’ generally, is said to have been first appropriated to mathematics by the Pythagoreans.
In saying that arithmetic began with Pythagoras we have to distinguish between the uses of that word then and now.Αριθμητικηwith the Greeks was distinguished fromλογιστικη, the science of calculation. It is the latter word which would cover arithmetic in our sense, or practical calculation; the termαριθμητικηwas restricted to the science of numbers considered in themselves, or, as we should say, the Theory of Numbers. Another way of putting the distinction was to say thatαριθμητικηdealt with absolute numbers or numbers in the abstract, andλογιστικηwith numberedthingsor concrete numbers; thusλογιστικηincluded simple problems about numbers of apples, bowls, or objects generally, such as are found in the Greek Anthology and sometimes involve simple algebraical equations.
The Theory of Numbers then began with Pythagoras (about572-497B. C.). It included definitions of the unit and of number, and the classification and definitions of the various classes of numbers, odd, even, prime, composite, and sub-divisions of these such as odd-even, even-times-even, &c. Again there were figured numbers, namely, triangular numbers, squares, oblong numbers, polygonal numbers (pentagons, hexagons, &c.) corresponding respectively to plane figures, and pyramidal numbers, cubes, parallelepipeds, &c., corresponding to solid figures in geometry. The treatment was mostly geometrical, the numbers being represented by dots filling up geometrical figures of the various kinds. The laws of formation of the various figured numbers were established. In this investigation thegnomonplayed an important part. Originally meaning the upright needle of a sun-dial, the term was next used for a figure like a carpenter’s square, and then was applied to a figure of that shape put round two sides of a square and making up a larger square. The arithmetical application of the term was similar. If we represent a unit by one dot and put round it three dots in such a way that the four form the corners of a square,threeis the first gnomon.Fivedots put at equal distances round two sides of the square containing four dots make up the next square (3²), andfiveis the second gnomon. Generally, if we haven²dots so arranged as to fill up a square withnfor its side, the gnomon to be put round it to make up the next square,(n+1)², has2n+1dots. In the formation of squares, therefore, the successive gnomons are the series of odd numbers following 1 (the first square), namely 3, 5, 7, ... In the formation ofoblongnumbers (numbers of the formn(n+1)), the first of which is 1. 2, the successive gnomons are the terms after 2 in the series ofevennumbers 2, 4, 6.... Triangular numbers are formed by adding to 1 (the first triangle) the terms after 1 in the series of natural numbers 1, 2, 3 ...; these are therefore the gnomons (by analogy) for triangles. The gnomons for pentagonal numbersare the terms after 1 in the arithmetical progression 1, 4, 7, 10 ... (with 3, or 5-2, as the common difference) and so on; the common difference of the successive gnomons for ana-gonal number isa-2.
From the series of gnomons for squares we easily deduce a formula for finding square numbers which are the sum of two squares. For, the gnomon2n+1being the difference between the successive squaresn²and(n+1)², we have only to make2n+1a square. Suppose that2n+1=m²; thereforen=½(m²-1), and{½(m²-1)}²+m²={½(m²+1)}², wheremis any odd number. This is the formula actually attributed to Pythagoras.
Pythagoras is said to have discovered the theory of proportionals or proportion. This was a numerical theory and therefore was applicable to commensurable magnitudes only; it was no doubt somewhat on the lines of Euclid, Book VII. Connected with the theory of proportion was that ofmeans, and Pythagoras was acquainted with three of these, the arithmetic, geometric, and sub-contrary (afterwards called harmonic). In particular Pythagoras is said to have introduced from Babylon into Greece the ‘most perfect’ proportion, namely:
a:(a+b)/2=2ab/(a+b):b,
where the second and third terms are respectively the arithmetic and harmonic mean betweenaandb. A particular case is 12:9=8:6.
This bears upon what was probably Pythagoras’s greatest discovery, namely that the musical intervals correspond to certain arithmetical ratios between lengths of string at the same tension, the octave corresponding to the ratio 2:1, the fifth to 3:2 and the fourth to 4:3. These ratios being the same as those of 12 to 6, 8, 9 respectively, we can understandhow the third term, 8, in the above proportion came to be called the ‘harmonic’ mean between 12 and 6.
The Pythagorean arithmetic as a whole, with the developments made after the time of Pythagoras himself, is mainly known to us through Nicomachus’sIntroductio arithmetica, Iamblichus’s commentary on the same, and Theon of Smyrna’s workExpositio rerum mathematicarum ad legendum Platonem utilium. The things in these books most deserving of notice are the following.
First, there is the description of a ‘perfect’ number (a number which is equal to the sum of all its parts, i.e. all its integral divisors including 1 but excluding the number itself), with a statement of the property that all such numbers end in 6 or 8. Four such numbers, namely 6, 28, 496, 8128, were known to Nicomachus. The law of formation for such numbers is first found in Eucl. IX. 36 proving that, if the sum(Sn)ofnterms of the series 1, 2, 2², 2³ ... is prime, thenSn.2n-1is a perfect number.
Secondly, Theon of Smyrna gives the law of formation of the series of ‘side-’ and ‘diameter-’ numbers which satisfy the equations2x²-y²=±1. The law depends on the proposition proved in Eucl. II. 10 to the effect that(2x+y)²-2(x+y)²=2x²-y², whence it follows that, ifx,ysatisfy either of the above equations, then2x+y,x+yis a solution in higher numbers of the other equation. The successive solutions give values fory/x, namely 1/1, 3/2, 7/5, 17/12, 41/29, ..., which are successive approximations to the value of √2 (the ratio of the diagonal of a square to its side). The occasion for this method of approximation to √2 (which can be carried as far as we please) was the discovery by the Pythagoreans of the incommensurable or irrational in this particular case.
Thirdly, Iamblichus mentions a discovery by Thymaridas, a Pythagorean not later than Plato’s time, called theεπανθημα(‘bloom’) of Thymaridas, and amounting to the solutionof any number of simultaneous equations of the following form:
x+x1+ x2+ ... + xn-1= s,x + x1= a1,x + x2= a2,....x+xn-1= an-1,
the solution beingx=((a1+a2+...+an-1)-s)/(n-2).
The rule is stated in general terms, but the above representation of its effect shows that it is a piece of pure algebra.
The Pythagorean contributions to geometry were even more remarkable. The most famous proposition attributed to Pythagoras himself is of course the theorem of Eucl. I. 47 that the square on the hypotenuse of any right-angled triangle is equal to the sum of the squares on the other two sides. But Proclus also attributes to him, besides the theory of proportionals, the construction of the ‘cosmic figures’, the five regular solids.
One of the said solids, the dodecahedron, has twelve regular pentagons for faces, and the construction of a regular pentagon involves the cutting of a straight line ‘in extreme and mean ratio’ (Eucl. II. 11 and VI. 30), which is a particular case of the method known as theapplication of areas. This method was fully worked out by the Pythagoreans and proved one of the most powerful in all Greek geometry. The most elementary case appears in Eucl. I. 44, 45, where it is shown how to apply to a given straight line as base a parallelogram with one angle equal to a given angle and equal in area to any given rectilineal figure; this construction is the geometrical equivalent of arithmeticaldivision. The general case is that in which the parallelogram, though applied to the straight line, overlaps it or falls short of it in such a way that the partof the parallelogram which extends beyond or falls short of the parallelogram of the same angle and breadth on the given straight line itself (exactly) as base is similar to any given parallelogram (Eucl. VI. 28, 29). This is the geometrical equivalent of the solution of the most general form of quadratic equationax±mx²=C, so far as it has real roots; the condition that the roots may be real was also worked out (=Eucl. VI. 27). It is in the form of ‘application of areas’ that Apollonius obtains the fundamental property of each of the conic sections, and, as we shall see, it is from the terminology of application of areas that Apollonius took the three namesparabola,hyperbola, andellipsewhich he was the first to give to the three curves.
Another problem solved by the Pythagoreans was that of drawing a rectilineal figure which shall be equal in area to one given rectilineal figure and similar to another. Plutarch mentions a doubt whether it was this problem or the theorem of Eucl. I. 47 on the strength of which Pythagoras was said to have sacrificed an ox.
The main particular applications of the theorem of the square on the hypotenuse, e. g. those in Euclid, Book II, were also Pythagorean; the construction of a square equal to a given rectangle (Eucl. II. 14) is one of them, and corresponds to the solution of the pure quadratic equationx²=ab.
The Pythagoreans knew the properties of parallels and proved the theorem that the sum of the three angles of any triangle is equal to two right angles.
As we have seen, the Pythagorean theory of proportion, being numerical, was inadequate in that it did not apply to incommensurable magnitudes; but, with this qualification, we may say that the Pythagorean geometry covered the bulk of the subject-matter of Books I, II, IV and VI of Euclid’sElements. The case is less clear with regard to Book III of theElements; but, as the main propositions of that Book were known to Hippocrates of Chios in the second half of the fifthcenturyB. C., we conclude that they, too, were part of the Pythagorean geometry.
Lastly, the Pythagoreans discovered the existence of the incommensurable or irrational in the particular case of the diagonal of a square in relation to its side. Aristotle mentions an ancient proof of the incommensurability of the diagonal with the side by areductio ad absurdumshowing that, if the diagonal were commensurable with the side, it would follow that one and the same number is both odd and even. This proof was doubtless Pythagorean.
A word should be added about the Pythagorean astronomy. Pythagoras was the first to hold that the earth (and no doubt each of the other heavenly bodies also) is spherical in shape, and he was aware that the sun, moon and planets have independent movements of their own in a sense opposite to that of the daily rotation; but he seems to have kept the earth in the centre. His successors in the school (one Hicetas of Syracuse and Philolaus are alternatively credited with this innovation) actually abandoned the geocentric idea and made the earth, like the sun, the moon, and the other planets, revolve in a circle round the ‘central fire’, in which resided the governing principle ordering and directing the movement of the universe.
The geometry of which we have so far spoken belongs to the Elements. But, before the body of the Elements was complete, the Greeks had advanced beyond the Elements. By the second half of the fifth centuryB. C.they had investigated three famous problems in higher geometry, (1) the squaring of the circle, (2) the trisection of any angle, (3) the duplication of the cube. The great names belonging to this period are Hippias of Elis, Hippocrates of Chios, and Democritus.
Hippias of Elis invented a certain curve described by combining two uniform movements (one angular and the other rectilinear) taking the same time to complete. Hippias himselfused his curve for the trisection of any angle or the division of it in any ratio; but it was afterwards employed by Dinostratus, a brother of Eudoxus’s pupil Menaechmus, and by Nicomedes for squaring the circle, whence it got the nameτετραγωνιζουσα,quadratrix.
Hippocrates of Chios is mentioned by Aristotle as an instance to prove that a man may be a distinguished geometer and, at the same time, a fool in the ordinary affairs of life. He occupies an important place both in elementary geometry and in relation to two of the higher problems above mentioned. He was, so far as is known, the first compiler of a book of Elements; and he was the first to prove the important theorem of Eucl. XII. 2 that circles are to one another as the squares on their diameters, from which he further deduced that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring oflunes, which was intended to lead up to the squaring of the circle. The essential portions of the tract are preserved in a passage of Simplicius’s commentary on Aristotle’sPhysics, which contains substantial extracts from Eudemus’s lostHistory of Geometry. Hippocrates showed how to square three particular lunes of different kinds and then, lastly, he squared the sum of a circle and a certain lune. Unfortunately the last-mentioned lune was not one of those which can be squared, so that the attempt to square the circle in this way failed after all.
Hippocrates also attacked the problem of doubling the cube. There are two versions of the origin of this famous problem. According to one story an old tragic poet had represented Minos as having been dissatisfied with the size of a cubical tomb erected for his son Glaucus and having told the architect to make it double the size while retaining the cubical form. The other story says that the Delians, suffering from a pestilence, consulted the oracle and were told todouble a certain altar as a means of staying the plague. Hippocrates did not indeed solve the problem of duplication, but reduced it to another, namely that of finding two mean proportionals in continued proportion between two given straight lines; and the problem was ever afterwards attacked in this form. Ifx,ybe the two required mean proportionals between two straight linesa,b, thena:x=x:y=y:b, whenceb/a=(x/a)³, and, as a particular case, ifb=2a,x³=2a³, so that, whenxis found, the cube is doubled.
Democritus wrote a large number of mathematical treatises, the titles only of which are preserved. We gather from one of these titles, ‘On irrational lines and solids’, that he wrote on irrationals. Democritus realized as fully as Zeno, and expressed with no less piquancy, the difficulty connected with the continuous and the infinitesimal. This appears from his dilemma about the circular base of a cone and a parallel section; the section which he means is a section ‘indefinitely near’ (as the phrase is) to the base, i. e. thevery nextsection, as we might say (if there were one). Is it, said Democritus, equal or not equal to the base? If it is equal, so will the very next section to it be, and so on, so that the cone will really be, not a cone, but a cylinder. If it is unequal to the base and in fact less, the surface of the cone will be jagged, like steps, which is very absurd. We may be sure that Democritus’s work on ‘The contact of a circle or a sphere’ discussed a like difficulty.
Lastly, Archimedes tells us that Democritus was the first to state, though he could not give a rigorous proof, that the volume of a cone or a pyramid is one-third of that of the cylinder or prism respectively on the same base and having equal height, theorems first proved by Eudoxus.
We come now to the time of Plato, and here the great names are Archytas, Theodoras of Cyrene, Theaetetus, and Eudoxus.
Archytas (about 430-360B. 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 ortorewith inner diameternil.
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 (427-347B. 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 planetsmay 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 theMeno(86E-87C). To Plato himself are attributed (1) a formula(n²-1)²+(2n)²=(n²+1)²for 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 (mathematicalanalysis 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 (408-355B. 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 equalnumbers, 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 circleas nearly as we please. The method of exhaustion used, for the purpose of proof byreductio 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 apparentmotions 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 thehippopede(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 388-315B. 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. Ifa:x=x:y=y:b, thenx²=ay,y²=bxandxy=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, andOn Risings andSettings. The former work is the earliest Greek textbook which has reached us intact. It was before Euclid when he wrote hisPhaenomena, and there are many points of contact between the two books.
Euclid flourished about 300B. C.or a little earlier. His great work, theElementsin 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 theDataandOn Divisions(of figures), the first of which survives in Greek and the second in Arabic only; also thePseudaria, now lost, which was a sort of guide to fallacies in geometrical reasoning. The treatises on higher geometry are all lost; they include (1) theConicsin four Books, which covered almost the same ground as the first three Books of Apollonius’sConics, 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 perpendiularto a generator in each case; (2) thePorismsin 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) theSurface-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’sConics.
In applied mathematics Euclid wrote (1) thePhaenomena, a work on spherical astronomy in whichὁ ὁριζων(withoutκυκλοςor any qualifying words) appears for the first time in the sense ofhorizon; (2) theOptics, a kind of elementary treatise on perspective: these two treatises are extant in Greek; (3) a work on the Elements of Music. TheSectio 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 310-230B. 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 andmoon. 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 287B. C.and was killed at the sack of Syracuse by Marcellus’s army in 212B. 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 shoutingἑυρηκα, 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 circumscribinga 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 CylinderI, II,Measurement of a Circle,On Conoids and Spheroids,On Spirals,On Plane EquilibriumsI, II, theSandreckoner,Quadrature of the Parabola,On Floating BodiesI, II, and lastly theMethod(only discovered in 1906). The difficult Cattle-Problem is also attributed to him, and aLiber Assumptorumwhich 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 theαρβηλος, shoemaker’s knife) are quite likely to be of Archimedean origin. Among lost works were theCatoptrica,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 realintegration; 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-discoveredMethodis especially interesting as showing how Archimedes originally obtained his results; this was by a clever mechanical method of (theoretically)weighinginfinitesimal 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 anavoidanceof 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 theMeasurement 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 callπ). 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.
ThePlane Equilibriumsis 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 AristotelianMechanicsmerely ‘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 gravityof 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.
TheSandreckoneris remarkable for the development in it of a system for expressing very large numbers byordersandperiodsbased 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 treatiseOn Floating Bodieswith 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 extraordinarytour 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 theMethod; thepreface 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. ThePlatonicusof 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 themost 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 workOn the Measurement of the EarthEratosthenes 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 aGeographicain 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, theConics. He is mentioned as a famous astronomer of the reign of Ptolemy Euergetes (247-222B. C.); and he dedicated the fourth and later Books of theConicsto King Attalus I of Pergamum (241-197B. C.).
TheConics, 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’sConics, 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 anexcess(ὑπερβολη) and application with adeficiency(ελλειψις), give the properties of the three curves respectively and account for the namesparabola,hyperbola, andellipse, 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 asminimumormaximumstraight 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 theevolutesof the three conics.
Several other works of Apollonius are described by Pappus as forming part of the ‘Treasury of Analysis’. All are lost except theSectio Rationisin 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 treatiseSectio Spatiidealt 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) OnDeterminate Section; (2)ContactsorTangencies, 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 satisfactorilyrestored); (3)Plane Loci, i. e. loci which are straight lines or circles; (4)Νευσεις,Inclinationes(the general problem called aνευσιςbeing to insert between two lines, straight or curved, a straight line of given lengthvergingto 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) aComparison 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 cochliasor cylindrical helix; (7) a ‘General Treatise’, which apparently dealt with the fundamental assumptions, &c., of elementary geometry; (8) a work onunordered 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 toπcloser than that in Archimedes’sMeasurement 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 calledε(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.
Nicomedes, probably intermediate in date between Eratosthenes and Apollonius, was the inventor of theconchoidorcochloid, of which, according to Pappus, there were three varieties. Diocles (about the end of the second centuryB. C.) is known as the discoverer of thecissoidwhich was used for duplicating the cube. He also wrote a bookπερι 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 atoreor anchor-ring.
Perseus is known as the discoverer and investigator of thespiric sections, i. e. certain sections of theσπειρα, one variety of which is thetore. Thespireis 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 onIsometric 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 centuryB. C.) wrote what became known as ‘Book XIV’ of theElementscontaining 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 135-51B. C.) wrote on geographyand astronomy under the titlesOn the Oceanandπερι 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 itlooks), 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 70B. 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 centuryA. D.) to revive interest in the subject. From the way in which, in his greatCollection, 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 hisCollectionremains, 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’sCollectionis an enunciationwhich 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 126B. 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.
TheSphaericaof Theodosius of Bithynia (written, say, 20B. C.) contains no trigonometry. It is otherwise with theSphaericaof 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, theSyntaxis, written aboutA. D.150and originally calledΜαθηματικη συνταξις, came to be known asΜεγαλη συνταξις; the Arabs made up from the superlativeμεγιστοςthe word al-Majisti which becameAlmagest.
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 360μοιραι, 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 103p53′ 23″). The Table of Chords is equivalent to a table of thesinesof the halves of the angles in the table, for, if (crd. 2α) represents the chord subtended by an angle of 2α(crd. 2α)/120 = sinα. 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 ½θ. 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, anOpticsin 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 centuryA. 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 ofDefinitionswhich hascome 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 (theGeometrica,Geodaesia,Stereometrica,Mensurae,Liber geëponicus) is genuine, we must now regard as more authoritative the genuineMetricadiscovered 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 triangleΔ= √{s(s-a) (s-b) (s-c)}, a formula now known to be due to Archimedes, (3) an allusion to limits to the value ofπfound by Archimedes and more exact than the 3-1/7 and 3-10/71 obtained in theMeasurement of a Circle.
Book I of theMetricacalculates 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’sMethod; 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’sDivisions(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 ourmethod, and theGeometricacontains two interesting indeterminate problems.
Heron also wrotePneumatica(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, andMechanics. TheMechanicshas 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 hisCollection.
We come, lastly, to Algebra. Problems involving simple equations are found in the Papyrus Rhind, in theEpanthemaof Thymaridas already referred to, and in the arithmetical epigrams in the Greek Anthology (Plato alludes to this class of problem in theLaws, 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 equationsx²+y²=z²and2x²-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. aboutA. D.250). He used (1) a sign for the unknown quantity, which he callsαριθμος, and compendia for its powers up to the sixth; (2) a sign (symbol-minus.png) with the effect of ourminus. The latter sign probably representsΛΙ, an abbreviation for the root of the wordλειπειν(to be wanting); the sign forαριθμος(symbol-arithmos.png) is most likely an abbreviation for the lettersαρ; the compendia for the powers of the unknown areΔΥforδυναμις, the square,ΚΥforκυβος, the cube, and so on. Diophantus shows that he solved quadratic equations by rule, like Heron. HisArithmetica, ofwhich 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 sumplusorminusany one of the numbers is a square’, ‘To find three numbers such that the product of any twoplusorminusthe 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 bookOn Polygonal Numbersof 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 centuryA. 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 theHistory of Geometryand theHistory of Astronomyby 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.