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 a _reductio ad absurdum_ showing 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 century B.

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 himself used 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 name tet?a???????sa {tetragonizousa}, _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 of _lunes_, which was intended to lead up to the squaring of the circle. The essential portions of the tract are preserved in a pa.s.sage of Simplicius's commentary on Aristotle's _Physics_, which contains substantial extracts from Eudemus's lost _History 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 to double 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. If _x_, _y_ be the two required mean proportionals between two straight lines _a_, _b_, then _a:x=x:y=y:b_, whence _b/a=(x/a)_, and, as a particular case, if _b=2a_, _x=2a_, so that, when _x_ is found, the cube is doubled.

Democritus wrote a large number of mathematical treatises, the t.i.tles only of which are preserved. We gather from one of these t.i.tles, '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. the _very next_ section, 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 430-360 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 v3, v5 ... v17 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 a.s.sumption of which the apparent motions of the planets 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 ill.u.s.trations, especially that in the second geometrical pa.s.sage in the _Meno_ (86E-87C). To Plato himself are attributed (1) a formula _(n-1)+(2n)=(n+1)_ for finding two square numbers the sum of which is a square number, (2) the invention of the method of a.n.a.lysis, which he is said to have explained to Leodamas of Thasos (_mathematical_ a.n.a.lysis 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 Weierstra.s.s repeats it word for word as his definition of equal 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 a.s.serted 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 circ.u.mference of the circle. Warned by the unanswerable arguments of Zeno against infinitesimals, mathematicians subst.i.tuted 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 a.s.signed magnitude of the same kind, however small): and this again depends on an a.s.sumption 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 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 _x=ay_, _y=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 pa.s.s 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 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 subst.i.tutes 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 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 a.s.sumed 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 ? ?????? {ho horizon} (without ??????

{kyklos} or any qualifying words) appears for the first time in the sense of _horizon_; (2) the _Optics_, a kind of elementary treatise on perspective: these two treatises are extant in Greek; (3) a work on the Elements of Music. The _Sectio Canonis_, which has come down under the name of Euclid, can, however, hardly be his in its present form.

In the period between Euclid and Archimedes comes Aristarchus of Samos (about 310-230 B. C.), famous for having antic.i.p.ated 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 cla.s.sical in form and style, it lays down certain hypotheses and then deduces therefrom, by rigorous geometry, the sizes and distances of the sun and moon. If the hypotheses had been exact, the results would have been correct too; but Aristarchus in fact a.s.sumed a certain angle to be 87 which is really 89 50', and the angle subtended at the centre of the earth by the diameter of either the sun or the moon to be 2, whereas we know from Archimedes that Aristarchus himself discovered that the latter angle is only 1/2. The effect of Aristarchus's geometry is to find arithmetical limits to the values of what are really trigonometrical ratios of certain small angles, namely

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

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

Archimedes was born about 287 B. C. and was killed at the sack of Syracuse by Marcellus's army in 212 B. C. The stories about him are well known, how he said 'Give me a place to stand on, and I will move the earth' (pa ? ?a? ???? ta? ?a? {pa bo kai kino tan gan}); how, having thought of the solution of the problem of the crown when in the bath, he ran home naked shouting ?????a, ?????a {heureka, heureka}; 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 circ.u.mscribing 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 a.s.sumptorum_ 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 a????? {arbelos}, 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 circ.u.mscribing 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 a.n.a.lytical equivalents are set down, seen to amount to real _integration_; this is so with his investigation of the areas of a parabolic segment and a spiral, the surface and volume of a sphere, and the volume of any segments of the conoids and spheroids.

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

In the _Measurement of a Circle_, after proving by exhaustion that the area of a circle is equal to a right-angled triangle with the perpendicular sides equal respectively to the radius and the circ.u.mference of the circle, Archimedes finds, by sheer calculation, upper and lower limits to the ratio of the circ.u.mference of a circle to its diameter (what we call p {p}). This he does by inscribing and circ.u.mscribing regular polygons of 96 sides and calculating approximately their respective perimeters. He begins by a.s.suming as known certain approximate values for v3, namely 1351/780 > v3 > 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 > p {p} > 3-10/71.

The _Plane Equilibriums_ is the first scientific treatise on the first principles of mechanics, which are established by pure geometry. The most important result established in 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', a.s.serting 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 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 (10,000). 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 a.s.sumption 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 a.s.sumption that bodies which are forced upwards in a fluid are forced upwards along the perpendiculars to the surface which pa.s.s 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 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.

[4] This word primarily means an all-round athlete, a winner in all five of the sports const.i.tuting the pe?ta???? {pentathlon}, namely jumping, discus-throwing, running, wrestling, and boxing (or javelin-throwing).

The most famous achievement of Eratosthenes was his measurement of the earth. Archimedes quotes an earlier measurement which made the circ.u.mference 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 circ.u.mference of the earth 250,000 stades, which Eratosthenes seems later, for some reason, to have changed to 252,000 stades. On the most probable a.s.sumption 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 pa.s.sed 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_ (pa?a??? {parabole}), application with an _excess_ (?pe???? {hyperbole}) and application with a _deficiency_ (e??e????

{elleipsis}), give the properties of the three curves respectively and account for the names _parabola_, _hyperbola_, and _ellipse_, by which Apollonius called them for the first time. The princ.i.p.al 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 const.i.tute 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 cla.s.ses 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 a.n.a.lysis'. 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'.