The most famous achievement of Eratosthenes was his measurement of the earth. Archimedes quotes an earlier measurement which made the circumference of the earth 300,000 stades. Eratosthenes improved upon this. He observed that at the summer solstice at Syene, at noon, the sun cast no shadow, while at the same moment the upright gnomon at Alexandria cast a shadow corresponding to an angle between the gnomon and the sun’s rays of 1/50th of four right angles. The distance between Syene and Alexandria being known to be 5,000 stades, this gave for the circumference of the earth 250,000 stades, which Eratosthenes seems later, for some reason, to have changed to 252,000 stades. On the most probable assumption as to the length of the stade used, the 252,000 stades give about 7,850 miles, only 50 miles less than the true polar diameter.
In the work On the Measurement of the Earth Eratosthenes is said to have discussed other astronomical matters, the distance of the tropic and polar circles, the sizes and distances of the sun and moon, total and partial eclipses, &c. Besides other works on astronomy and chronology, Eratosthenes wrote a Geographica in three books, in which he first gave a history of geography up to date and then passed on to mathematical geography, the spherical shape of the earth, &c., &c.
Apollonius of Perga was with justice called by his contemporaries the ‘Great Geometer’, on the strength of his great treatise, the Conics. He is mentioned as a famous astronomer of the reign of Ptolemy Euergetes (247-222 B. C.); and he dedicated the fourth and later Books of the Conics to King Attalus I of Pergamum (241-197 B. C.).
The Conics, a colossal work, originally in eight Books, survives as to the first four Books in Greek and as to three more in Arabic, the eighth being lost. From Apollonius’s prefaces we can judge of the relation of his work to Euclid’s Conics, the content of which answered to the first three Books of Apollonius. Although Euclid knew that an ellipse could be otherwise produced, e. g. as an oblique section of a right cylinder, there is no doubt that he produced all three conics from right cones like his predecessors. Apollonius, however, obtains them in the most general way by cutting any oblique cone, and his original axes of reference, a diameter and the tangent at its extremity, are in general oblique; the fundamental properties are found with reference to these axes by ‘application of areas’, the three varieties of which, application (παραβολη), application with an excess (ὑπερβολη) and application with a deficiency (ελλειψις), give the properties of the three curves respectively and account for the names parabola, hyperbola, and ellipse, by which Apollonius called them for the first time. The principal axes only appear, as a particular case, after it has been shown that the curves have a like property when referred to any other diameter and the tangent at its extremity, instead of those arising out of the original construction. The first four Books constitute what Apollonius calls an elementary introduction; the remaining Books are specialized investigations, the most important being Book V (on normals) and Book VII (mainly on conjugate diameters). Normals are treated, not in connexion with tangents, but as minimum or maximum straight lines drawn to the curves from different points or classes of points. Apollonius discusses such questions as the number of normals that can be drawn from one point (according to its position) and the construction of all such normals. Certain propositions of great difficulty enable us to deduce quite easily the Cartesian equations to the evolutes of the three conics.
Several other works of Apollonius are described by Pappus as forming part of the ‘Treasury of Analysis’. All are lost except the Sectio Rationis in two Books, which survives in Arabic and was published in a Latin translation by Halley in 1706. It deals with all possible cases of the general problem ‘given two straight lines either parallel or intersecting, and a fixed point on each, to draw through any given point a straight line which shall cut off intercepts from the two lines (measured from the fixed points) bearing a given ratio to one another’. The lost treatise Sectio Spatii dealt similarly with the like problem in which the intercepts cut off have to contain a given rectangle.
The other treatises included in Pappus’s account are (1) On Determinate Section; (2) Contacts or Tangencies, Book II of which is entirely devoted to the problem of drawing a circle to touch three given circles (Apollonius’s solution can, with the aid of Pappus’s auxiliary propositions, be satisfactorily restored); (3) Plane Loci, i. e. loci which are straight lines or circles; (4) Νευσεις, Inclinationes (the general problem called a νευσις being to insert between two lines, straight or curved, a straight line of given length verging to a given point, i. e. so that, if produced, it passes through the point, Apollonius restricted himself to cases which could be solved by ‘plane’ methods, i. e. by the straight line and circle only).
Apollonius is also said to have written (5) a Comparison of the dodecahedron with the icosahedron (inscribed in the same sphere), in which he proved that their surfaces are in the same ratio as their volumes; (6) On the cochlias or cylindrical helix; (7) a ‘General Treatise’, which apparently dealt with the fundamental assumptions, &c., of elementary geometry; (8) a work on unordered irrationals, i. e. irrationals of more complicated form than those of Eucl. Book X; (9) On the burning-mirror, dealing with spherical mirrors and probably with mirrors of parabolic section also; (10) ωκυτοκιον (‘quick delivery’). In the last-named work Apollonius found an approximation to π closer than that in Archimedes’s Measurement of a Circle; and possibly the book also contained Apollonius’s exposition of his notation for large numbers according to ‘tetrads’ (successive powers of the myriad).
In astronomy Apollonius is said to have made special researches regarding the moon, and to have been called ε (Epsilon) because the form of that letter is associated with the moon. He was also a master of the theory of epicycles and eccentrics.
With Archimedes and Apollonius Greek geometry reached its culminating point; indeed, without some more elastic notation and machinery such as algebra provides, geometry was practically at the end of its resources. For some time, however, there were capable geometers who kept up the tradition, filling in details, devising alternative solutions of problems, or discovering new curves for use or investigation.
Nicomedes, probably intermediate in date between Eratosthenes and Apollonius, was the inventor of the conchoid or cochloid, of which, according to Pappus, there were three varieties. Diocles (about the end of the second century B. C.) is known as the discoverer of the cissoid which was used for duplicating the cube. He also wrote a book περι πυρειων, On burning-mirrors, which probably discussed, among other forms of mirror, surfaces of parabolic or elliptic section, and used the focal properties of the two conics; it was in this work that Diocles gave an independent and clever solution (by means of an ellipse and a rectangular hyperbola) of Archimedes’s problem of cutting a sphere into two segments in a given ratio. Dionysodorus gave a solution by means of conics of the auxiliary cubic equation to which Archimedes reduced this problem; he also found the solid content of a tore or anchor-ring.