[3] In the case of the parabola, the base (as distinct from the 'erect side') of the rectangle is what is called the _abscissa_ (Gk.

ap?te??e?? {apotemnomene}, 'cut off') of the ordinate, and the rectangle itself is equal to the square on the ordinate. In the case of the central conics, the base of the rectangle is 'the transverse side of the figure' or the transverse diameter (the diameter of reference), and the rectangle is equal to the square on the diameter conjugate to the diameter of reference.

Again, if we take up a textbook of geometry written in accordance with the most modern Education Board circular or University syllabus, we shall find that the phraseology used (except where made more colloquial and less scientific) is almost all pure Greek. The Greek tongue was extraordinarily well adapted as a vehicle of scientific thought. One of the characteristics of Euclid's language which his commentator Proclus is most fond of emphasizing is its marvellous _exactness_ (a???e?a {akribeia}). The language of the Greek geometers is also wonderfully concise, notwithstanding all appearances to the contrary. One of the complaints often made against Euclid is that he is 'diffuse'. Yet (apart from abbreviations in writing) it will be found that the exposition of corresponding matters in modern elementary textbooks generally takes up, not less, but more s.p.a.ce. And, to say nothing of the perfect finish of Archimedes's treatises, we shall find in Heron, Ptolemy and Pappus veritable models of concise statement. The purely geometrical proof by Heron of the formula for the area of a triangle, ?{D}=v_{s(s-a)(s-b)(s-c)}_, and the geometrical propositions in Book I of Ptolemy's _Syntaxis_ (including 'Ptolemy's Theorem') are cases in point.

The principles of geometry and arithmetic (in the sense of the theory of numbers) are stated in the preliminary matter of Books I and VII of Euclid. But Euclid was not their discoverer; they were gradually evolved from the time of Pythagoras onwards. Aristotle is clear about the nature of the principles and their cla.s.sification. Every demonstrative science, he says, has to do with three things, the subject-matter, the things proved, and the things from which the proof starts (e? ?? {ex hon}). It is not everything that can be proved, otherwise the chain of proof would be endless; you must begin somewhere, and you must start with things admitted but indemonstrable. These are, first, principles common to all sciences which are called _axioms_ or _common opinions_, as that 'of two contradictories one must be true', or 'if equals be subtracted from equals, the remainders are equal'; secondly, principles peculiar to the subject-matter of the particular science, say geometry. First among the latter principles are definitions; there must be agreement as to what we mean by certain terms. But a definition a.s.serts nothing about the existence or non-existence of the thing defined. The existence of the various things defined has to be _proved_ except in the case of a few primary things in each science the existence of which is indemonstrable and must be _a.s.sumed_ among the first principles of the science; thus in geometry we must a.s.sume the existence of points and lines, and in arithmetic of the unit. Lastly, we must a.s.sume certain other things which are less obvious and cannot be proved but yet have to be accepted; these are called _postulates_, because they make a demand on the faith of the learner. Euclid's Postulates are of this kind, especially that known as the parallel-postulate.

The methods of solution of problems were no doubt first applied in particular cases and then gradually systematized; the technical terms for them were probably invented later, after the methods themselves had become established.

One method of solution was the _reduction_ of one problem to another.

This was called apa???? {apagoge}, a term which seems to occur first in Aristotle. But instances of such reduction occurred long before.

Hippocrates of Chios reduced the problem of duplicating the cube to that of finding two mean proportionals in continued proportion between two straight lines, that is, he showed that, if the latter problem could be solved, the former was thereby solved also; and it is probable that there were still earlier cases in the Pythagorean geometry.

Next there is the method of mathematical _a.n.a.lysis_. This method is said to have been 'communicated' or 'explained' by Plato to Leodamas of Thasos; but, like reduction (to which it is closely akin), a.n.a.lysis in the mathematical sense must have been in use much earlier. _a.n.a.lysis_ and its correlative _synthesis_ are defined by Pappus: 'in a.n.a.lysis we a.s.sume that which is sought as if it were already done, and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the cla.s.s of principles.

But in synthesis, reversing the process, we take as already done that which was last arrived at in the a.n.a.lysis, and, by arranging in their natural order as consequences what were before antecedents and successively connecting them one with another, we arrive finally at the construction of that which was sought.'

The method of _reductio ad absurdum_ is a variety of a.n.a.lysis. Starting from a hypothesis, namely the contradictory of what we desire to prove, we use the same process of a.n.a.lysis, carrying it back until we arrive at something admittedly false or absurd. Aristotle describes this method in various ways as _reductio ad absurdum_, proof _per impossibile_, or proof leading to the impossible. But here again, though the term was new, the method was not. The paradoxes of Zeno are cla.s.sical 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 a.s.signed, (1) the _enunciation_ (p??tas?? {protasis}), (2) the _setting-out_ (e??es??

{ekthesis}), (3) the d????s?? {diorismos}, being a re-statement of what we are required to do or prove, not in general terms (as in the _enunciation_), but with reference to the particular data contained in the _setting-out_, (4) the _construction_ (?atas?e?? {kataskeue}), (5) the _proof_ (ap?de???? {apodeixis}), (6) the _conclusion_ (s?pe?asa {symperasma}). 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 const.i.tuent part in the proposition, namely the statement of the conditions or limits of possibility, which was called by the same name d????s?? {diorismos}, definition or delimitation, as that applied to the third const.i.tuent 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 a.s.sert themselves even in their 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 ill.u.s.trated 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 ill.u.s.trates 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 superst.i.tion, 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 624-547 B. 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 a.s.sessment 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 1700 B. C.) and other doc.u.ments that Egyptian geometry consisted 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-qe?_, 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 ill.u.s.trate 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 a _theory_ of geometry. On this point we cannot do better than quote some remarks from Kant's preface to the second edition of his _Kritik 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 a _science_. 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 a _revolution_ brought 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 585 B. C. Now the Babylonians, as the result of observations continued through centuries, had discovered the period of 223 lunations after which eclipses recur. 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 works _On the Equinox_ and _On 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 word a??ata {mathemata}, 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. ?????t??? {Arithmetike} with the Greeks was distinguished from ????st??? {logistike}, the science of calculation. It is the latter word which would cover arithmetic in our sense, or practical calculation; the term a????t???

{arithmetike} was 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 that a????t??? {arithmetike} dealt with absolute numbers or numbers in the abstract, and ????st???

{logistike} with numbered _things_ or concrete numbers; thus ????st???

{logistike} included 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 (about 572-497 B. C.).

It included definitions of the unit and of number, and the cla.s.sification and definitions of the various cla.s.ses 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 the _gnomon_ played 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, _three_ is the first gnomon. _Five_ dots put at equal distances round two sides of the square containing four dots make up the next square (3), and _five_ is the second gnomon. Generally, if we have _n_ dots so arranged as to fill up a square with _n_ for its side, the gnomon to be put round it to make up the next square, _(n+1)_, has _2n+1_ dots. 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 of _oblong_ numbers (numbers of the form _n(n+1)_), the first of which is 1. 2, the successive gnomons are the terms after 2 in the series of _even_ numbers 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 a.n.a.logy) for triangles.

The gnomons for pentagonal numbers are 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 an _a_-gonal number is _a-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 gnomon _2n+1_ being the difference between the successive squares _n_ and _(n+1)_, we have only to make _2n+1_ a square. Suppose that _2n+1=m_; therefore _n=(m-1)_, and _{(m-1)}+m={(m+1)}_, where _m_ is 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 of _means_, 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 between _a_ and _b_. 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 understand how 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's _Introductio arithmetica_, Iamblichus's commentary on the same, and Theon of Smyrna's work _Expositio 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 (S_{n}) of _n_ terms of the series 1, 2, 2, 2 ... is prime, then S_{n}.2^{n-1} is a perfect number.

Secondly, Theon of Smyrna gives the law of formation of the series of 'side-' and 'diameter-' numbers which satisfy the equations _2x-y=1_.

The law depends on the proposition proved in Eucl. II. 10 to the effect that _(2x+y)-2(x+y)=2x-y_, whence it follows that, if _x_, _y_ satisfy either of the above equations, then _2x+y_, _x+y_ is a solution in higher numbers of the other equation. The successive solutions give values for _y/x_, namely 1/1, 3/2, 7/5, 17/12, 41/29, ..., which are successive approximations to the value of v2 (the ratio of the diagonal of a square to its side). The occasion for this method of approximation to v2 (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 the epa???a {epanthema} ('bloom') of Thymaridas, and amounting to the solution of any number of simultaneous equations of the following form:

_x+x1+x2+ ... +x_{n-1} = s_, _x+x1 = a1_, _x+x2 = a2_, ...

_x+x_{n-1} = a_{n-1}_,

the solution being _x=((a1+a2+ ... +a_{n-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 the _application 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 arithmetical _division_. 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 part of 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 equation _axmx=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 names _parabola_, _hyperbola_, and _ellipse_ which 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 equation _x=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's _Elements_. The case is less clear with regard to Book III of the _Elements_; but, as the main propositions of that Book were known to Hippocrates of Chios in the second half of the fifth century B. C., we conclude that they, too, were part of the Pythagorean geometry.