Part 6 (1/2)
The lost treatise _Sectio Spatii_ dealt similarly with the like problem in which the intercepts cut off have to contain a given rectangle.
The other treatises included in Pappus's account are (1) On _Determinate Section_; (2) _Contacts_ or _Tangencies_, Book II of which is entirely devoted to the problem of drawing a circle to touch three given circles (Apollonius's solution can, with the aid of Pappus's auxiliary propositions, be satisfactorily restored); (3) _Plane Loci_, i. e. loci which are straight lines or circles; (4) ?e?se?? {Neuseis}, _Inclinationes_ (the general problem called a ?e?s?? {neusis} being to insert between two lines, straight or curved, a straight line of given length _verging_ to a given point, i. e. so that, if produced, it pa.s.ses through the point, Apollonius restricted himself to cases which could be solved by 'plane' methods, i. e. by the straight line and circle only).
Apollonius is also said to have written (5) a _Comparison of the dodecahedron with the icosahedron_ (inscribed in the same sphere), in which he proved that their surfaces are in the same ratio as their volumes; (6) _On the cochlias_ or cylindrical helix; (7) a 'General Treatise', which apparently dealt with the fundamental a.s.sumptions, &c., of elementary geometry; (8) a work on _unordered irrationals_, i. e.
irrationals of more complicated form than those of Eucl. Book X; (9) _On the burning-mirror_, dealing with spherical mirrors and probably with mirrors of parabolic section also; (10) ???t????? {okytokion} ('quick delivery'). In the last-named work Apollonius found an approximation to p {p} closer than that in Archimedes's _Measurement of a Circle_; and possibly the book also contained Apollonius's exposition of his notation for large numbers according to 'tetrads' (successive powers of the myriad).
In astronomy Apollonius is said to have made special researches regarding the moon, and to have been called e {e} (Epsilon) because the form of that letter is a.s.sociated 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 the _conchoid_ or _cochloid_, of which, according to Pappus, there were three varieties. Diocles (about the end of the second century B. C.) is known as the discoverer of the _cissoid_ which was used for duplicating the cube. He also wrote a book pe??
p??e??? {peri pyreion}, _On burning-mirrors_, which probably discussed, among other forms of mirror, surfaces of parabolic or elliptic section, and used the focal properties of the two conics; it was in this work that Diocles gave an independent and clever solution (by means of an ellipse and a rectangular hyperbola) of Archimedes's problem of cutting a sphere into two segments in a given ratio. Dionysodorus gave a solution by means of conics of the auxiliary cubic equation to which Archimedes reduced this problem; he also found the solid content of a _tore_ or anchor-ring.
Perseus is known as the discoverer and investigator of the _spiric sections_, i. e. certain sections of the spe??a {speira}, one variety of which is the _tore_. The _spire_ is generated by the revolution of a circle about a straight line in its plane, which straight line may either be external to the circle (in which case the figure produced is the tore), or may cut or touch the circle.
Zenodorus was the author of a treatise on _Isometric figures_, the problem in which was to compare the content of different figures, plane or solid, having equal contours or surfaces respectively.
Hypsicles (second half of second century B. C.) wrote what became known as 'Book XIV' of the _Elements_ containing supplementary propositions on the regular solids (partly drawn from Aristaeus and Apollonius); he seems also to have written on polygonal numbers. A mediocre astronomical work (??af?????? {Anaphorikos}) attributed to him is the first Greek book in which we find the division of the zodiac circle into 360 parts or degrees.
Posidonius the Stoic (about 135-51 B. C.) wrote on geography and astronomy under the t.i.tles _On the Ocean_ and pe?? ete???? {peri meteoron}. He made a new but faulty calculation of the circ.u.mference of the earth (240,000 stades). _Per contra_, in a separate tract on the size of the sun (in refutation of the Epicurean view that it is as big as it _looks_), he made a.s.sumptions (partly guesswork) which give for the diameter of the sun a figure of 3,000,000 stades (39-1/4 times the diameter of the earth), a result much nearer the truth than those obtained by Aristarchus, Hipparchus, and Ptolemy. In elementary geometry Posidonius gave certain definitions (notably of parallels, based on the idea of equidistance).
Geminus of Rhodes, a pupil of Posidonius, wrote (about 70 B. C.) an encyclopaedic work on the cla.s.sification and content of mathematics, including the history of each subject, from which Proclus and others have preserved notable extracts. An-Nairizi (an Arabian commentator on Euclid) reproduces an attempt by one 'Aganis', who appears to be Geminus, to prove the parallel-postulate.
But from this time onwards the study of higher geometry (except sphaeric) seems to have languished, until that admirable mathematician, Pappus, arose (towards the end of the third century A. D.) to revive interest in the subject. From the way in which, in his great _Collection_, Pappus thinks it necessary to describe in detail the contents of the cla.s.sical works belonging to the 'Treasury of a.n.a.lysis'
we gather that by his time many of them had been lost or forgotten, and that he aimed at nothing less than re-establis.h.i.+ng geometry at its former level. No one could have been better qualified for the task.
Presumably such interest as Pappus was able to arouse soon flickered out; but his _Collection_ remains, after the original works of the great mathematicians, the most comprehensive and valuable of all our sources, being a handbook or guide to Greek geometry and covering practically the whole field. Among the original things in Pappus's _Collection_ is an enunciation which amounts to an antic.i.p.ation of what is known as Guldin's Theorem.
It remains to speak of three subjects, trigonometry (represented by Hipparchus, Menelaus, and Ptolemy), mensuration (in Heron of Alexandria), and algebra (Diophantus).
Although, in a sense, the beginnings of trigonometry go back to Archimedes (_Measurement of a Circle_), Hipparchus was the first person who can be proved to have used trigonometry systematically. Hipparchus, the greatest astronomer of antiquity, whose observations were made between 161 and 126 B. C., discovered the precession of the equinoxes, calculated the mean lunar month at 29 days, 12 hours, 44 minutes, 2-1/2 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 lat.i.tude and longitude (in relation to the ecliptic). He wrote a treatise in twelve Books on Chords in a Circle, equivalent to a table of trigonometrical sines. For calculating arcs in astronomy from other arcs given by means of tables he used propositions in spherical trigonometry.
The _Sphaerica_ of Theodosius of Bithynia (written, say, 20 B. C.) contains no trigonometry. It is otherwise with the _Sphaerica_ of Menelaus (fl. A. D. 100) extant in Arabic; Book I of this work contains propositions about spherical triangles corresponding to the main propositions of Euclid about plane triangles (e.g. congruence theorems and the proposition that in a spherical triangle the three angles are together greater than two right angles), while Book III contains genuine spherical trigonometry, consisting of 'Menelaus's Theorem' with reference to the sphere and deductions therefrom.
Ptolemy's great work, the _Syntaxis_, written about A. D. 150 and originally called ?a??at??? s??ta??? {Mathematike syntaxis}, came to be known as ?e?a?? s??ta??? {Megale syntaxis}; the Arabs made up from the superlative e??st?? {megistos} the word al-Majisti which became _Almagest_.
Book I, containing the necessary preliminaries to the study of the Ptolemaic system, gives a Table of Chords in a circle subtended by angles at the centre of increasing by half-degrees to 180. The circle is divided into 360 ???a? {moirai}, parts or degrees, and the diameter into 120 parts (t?ata {tmemata}); the chords are given in terms of the latter with s.e.xagesimal fractions (e. g. the chord subtended by an angle of 120 is 103^{p} 53' 23?). The Table of Chords is equivalent to a table of the _sines_ of the halves of the angles in the table, for, if (crd. 2 a {a}) represents the chord subtended by an angle of 2 a {a} (crd. 2 a {a})/120 = sin a {a}. 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 (?-f) {th-ph}, the third to that for sin ? {th}.
From (crd. 72) and (crd. 60) Ptolemy, by using these propositions successively, deduces (crd. 1) and (crd. ), 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 (?+f) {th+ph}.
Ptolemy wrote other minor astronomical works, most of which survive in Greek or Arabic, an _Optics_ in five Books (four Books almost complete were translated into Latin in the twelfth century), and an attempted proof of the parallel-postulate which is reproduced by Proclus.
Heron of Alexandria (date uncertain; he may have lived as late as the third century A. D.) was an almost encyclopaedic writer on mathematical and physical subjects. He aimed at practical utility rather than theoretical completeness; hence, apart from the interesting collection of _Definitions_ which has come down under his name, and his commentary on Euclid which is represented only by extracts in Proclus and an-Nairizi, his geometry is mostly mensuration in the shape of numerical examples worked out. As these could be indefinitely multiplied, there was a temptation to add to them and to use Heron's name. However much of the separate works edited by Hultsch (the _Geometrica_, _Geodaesia_, _Stereometrica_, _Mensurae_, _Liber geeponicus_) is genuine, we must now regard as more authoritative the genuine _Metrica_ discovered at Constantinople in 1896 and edited by H. Schone in 1903 (Teubner). Book I on the measurement of areas is specially interesting for (1) its statement of the formula used by Heron for finding approximations to surds, (2) the elegant geometrical proof of the formula for the area of a triangle ? {D} = v{_s (s-a) (s-b) (s-c)}, a formula now known to be due to Archimedes, (3) an allusion to limits to the value of p {p} found by Archimedes and more exact than the 3-1/7 and 3-10/71 obtained in the _Measurement of a Circle_.
Book I of the _Metrica_ calculates the areas of triangles, quadrilaterals, the regular polygons up to the dodecagon (the areas even of the heptagon, enneagon, and hendecagon are approximately evaluated), the circle and a segment of it, the ellipse, a parabolic segment, and the surfaces of a cylinder, a right cone, a sphere and a segment thereof. Book II deals with the measurement of solids, the cylinder, prisms, pyramids and cones and frusta thereof, the sphere and a segment of it, the anchor-ring or tore, the five regular solids, and finally the two special solids of Archimedes's _Method_; full use is made of all Archimedes's results. Book III is on the division of figures. The plane portion is much on the lines of Euclid's _Divisions_ (of figures). The solids divided in given ratios are the sphere, the pyramid, the cone and a frustum thereof. Incidentally Heron shows how he obtained an approximation to the cube root of a non-cube number (100). Quadratic equations are solved by Heron by a regular rule not unlike our method, and the _Geometrica_ contains two interesting indeterminate problems.
Heron also wrote _Pneumatica_ (where the reader will find such things as siphons, Heron's Fountain, penny-in-the-slot machines, a fire-engine, a water-organ, and many arrangements employing the force of steam), _Automaton-making_, _Belopoeca_ (on engines of war), _Catoptrica_, and _Mechanics_. The _Mechanics_ has been edited from the Arabic; it is (except for considerable fragments) lost in Greek. It deals with the puzzle of 'Aristotle's Wheel', the parallelogram of velocities, definitions of, and problems on, the centre of gravity, the distribution of weights between several supports, the five mechanical powers, mechanics in daily life (queries and answers). Pappus covers much the same ground in Book VIII of his _Collection_.
We come, lastly, to Algebra. Problems involving simple equations are found in the Papyrus Rhind, in the _Epanthema_ of Thymaridas already referred to, and in the arithmetical epigrams in the Greek Anthology (Plato alludes to this cla.s.s of problem in the _Laws_, 819 B, C); the Anthology even includes two cases of indeterminate equations of the first degree. The Pythagoreans gave general solutions in rational numbers of the equations _x+y=z_ and _2x-y=1_, which are indeterminate equations of the second degree.