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is lost but we have his work on Spherical Triangles, a learned performance, in which we find the construction of these triangles, and the trigonometrical method of resolving them in most cases necessary in the practice of ancient astronomy.
There is also another geometrical theory, perspective, with which it is doubted whether the ancients were acquainted. For my own part, I do not see how it can be questioned with regard to linear perspective: for this science, if we can give it the separate name, is nothing more than a very simple and easy application of the theory of similar triangles. In fact it consists only in representing on a plane, or given superficies, an object as it would appear when seen from a given point: or, in geometrical language, in projecting on a given surface the parts of an object by means of lines drawn from a fixed and given point to every point of the object. Now is not such a problem more than virtually contained in the Elements of Euclid? not to mention, that perhaps it has been explicitly solved in some of the works, which have not reached us. If, however, any one should not be satisfied with this indirect proof, I will produce him a direct one, taken from Vitruvius. The passage that includes it has not been translated quite agreeably to the true sense by Claude Perrault, and I cannot avoid adopting in preference the following translation given by Mr. Jalabert in the Memoirs of the Academy of Belles Lettres, vol. xxIII, p. 341.
'Agatharchus was the first who painted decorations for the theatre, which was at the time when Eschylus exhibited tragedies at Athens....... From
his example Democritus and Anaxagoras wrote on the subject, how, a point being fixed in a certain spot with regard to the eye and visual rays, certain lines proportionate to the natural distances were made to answer to it, in such a manner, that from a thing concealed, or which it would be difficult to guess at, images resembling objects arise; so that, for example, they represent buildings on the stage, which, though painted on a flat surface, appear to project in certain parts.' Vitr. book VII, pref.
Here, it seems to me, linear perspective is plainly described.
With regard to aerial perspective, which depends on the opposition and gradation of colours, the question is not so easily solved. Some moderns assert, that the ancients had only imperfect notions, founded on a kind of customary practice: but I confess the reasons adduced by count Caylus, in support of the opposite opinion, have great weight with me. Let the reader weigh with attention the following passage, extracted from a dissertation, in which that learned critic discusses this subject.
"Ancient painting, at least the most perfect and finished, no longer exists, to convince us how far the ancients had carried the art of perspective. It is certain, that even in the time of Augustus the works of Zeuxis, Protogenes, and the other great painters of the most flourishing era of Greece, were scarcely distinguishable, so much had the colours flown off and become effaced, and so wormeaten was the wood for portable pictures were painted on no other material, at least as far as we can learn from any historian.
historian. What now remains therefore, on which we can found our judgment, either for or against it? A few paintings on walls, which we may think ourselves happy in possessing, but which our taste for the antique should not lead us equally to admire. Beautiful as they are in certain respects, it is unquestionable, that they are not to be compared with those superb paintings, on which ancient writers have passed such high encomiums, while speaking to men who admired them as well as themselves, to men who felt all the merit of those masterpieces of sculpture, with regard to which we cannot suspect these authors of partiality, since we are able to judge of them ourselves, admire them daily, and know that they were both equally designed for the decoration of their temples and their public edifices. These arts follow each other: on this I must continue to insist, and I shall add, it is naturally impossible that the one, sculpture, should be elegant and sublime; while the other, painting, was reduced to a degree of insipidity and imperfection, such as must be the case with a picture destitute of relief, without any gradation of colour, in short void of what we call skill in harmony.' Mém. de l'Acad. des Belles Let. vol. XXIII, p. 323.
Were I writing a minute history of mathematics, I might give an ample list of the geometricians, who flourished from the time of Archimedes to the destruction of the Alexandrian school. I should quote Conon and Dositheus, both very learned, and both friends of Archimedes; Geminus, a mathematician of Rhodes, who wrote a work entitled
Enarrationes Geometria; &c.: but I shall confine myself here to giving the reader a succinct view of those, some of whose works have come down to us, and of whom therefore we can speak with some knowledge, without being wholly led by the mere narratives. of historians.
Theodosius is the first who presents himself with his treatise on Spherics, in which he examines the properties, which circles formed by cutting a sphere in all directions have with respect to each other. A. c. 60. This work, excellent in itself, may be considered as an introduction to spherical trigonometry. Most of the author's propositions now appear evident. at the first view: but, faithful to the maxims of the ancients, he demonstrates every thing with the greatest strictness, and with much elegance. We have likewise two other treatises by Theodosius, entitled, Of Habitations, and Of Days and Nights, which contain an explanation of the celestial phenomena, to be perceived in different parts of the Earth.
After Theodosius we proceed for three or four hundred years, without meeting with any geometrician of eminence, except Menelaus, who has already been mentioned. At length we come to Pappus and Diocles, of both of whom I have likewise spoken. with eulogium on occasion of the two particular problems of the duplication of the cube, and the trisection of an angle, and who reappear in this place on other accounts. A. D. 385. We shall meet with some other geometricians also of distinguished merit.
The mathematical collections of Pappus exhibit one of the most valuable monuments of ancient geometry.
ometry. In them the author has assembled together the substance of a great number of excellent works, almost all of which are now lost; and to these he has added several new, curious, and learned propositions of his own invention. This collection therefore is not to be considered as an ordinary compilation; though even in this view it would deserve very high esteem, as it gives us almost a complete view of the state of ancient mathematics. It was divided into eight books: the first two are lost; the subjects of the others are, in general, questions in geometry, with a few in astronomy and mechanics.
Among other researches Pappus proposed to him. self the problem of geometrical loci in all it's extent, and advanced a great way in it's solution. As it's completion required the assistance of algebra, I shall defer entering into it, till I come to the geometrical discoveries of des Cartes, in the third period.
Pappus gave the solution of another very curious problem, of a kind which at that time was absolutely novel it was that of finding on the superficies of a sphere spaces capable of being squared. By means of the theorems of Archimedes he demonstrates, that,
if a movable point, setting out from the summit of a hemisphere, traverse a quarter of the circumference, while this quarter of the circumference makes a complete revolution round the axis, the space, comprised between the circumference of the base and the spiral of double curvature described on the spherical surface by the movable point, is equal to the square of the diameter.' The proposition may easily be generalised; and we find, that, all the other cir