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Archimedes was a lover of glory; not of that vain phantom, which mediocrity pursues, yet cannot reach; but of solid glory, of that reputation, of that respect, which are due to the man of genius, who enlarges the limits of science. He desired, when he was dying, that a sphere inscribed in a cylinder might be engraved on his tomb, to perpetuate the memory of his most brilliant discovery. His desire was obeyed, but the Sicilians, his countrymen, having their minds turned on objects very different from geometry, soon forgot the man, who was their chief honour in the eyes of posterity. Two hundred years after his death, Cicero, being then quæstor in Sicily, gave Archimedes to the light, as he himself expresses it, a second time. Unable to learn from the Sicilians the place of his tomb, he sought for it by the symbol, which I have just mentioned, and six verses in greek inscribed on it's base. After much labour it was at length discovered overgrown with thorns in a field near Syracuse; and the Sicilians blushed for their ignorance and ingratitude.
Fifty years had scarcely elapsed after the death of Archimedes, when another geometrician appeared, by whom he was almost equalled, and who is at least beyond dispute the second of the ancient geometricians. A. C. 200. This was Apollonius, born at Perga in Pamphilia, whence he is called Apollonius Pergæus. His contemporaries styled him the great geometrician; and posterity has confirmed this honourable title, without detracting from the merit of Archimedes, to whom it assigns the first rank.
Apollonius had composed a great number of works on the higher geometry of his times, most of which are lost, or exist only in fragments; but we have almost the whole of his treatise on Conic Sections, which alone is sufficient to justify the high reputation of it's author. This treatise was divided into eight books. The first four have reached us in their original language, the greek: the following three have been preserved only in an arabic translation, made about the year 1250, and rendered into latin in the middle of the seventeenth century: the eighth was entirely lost. The celebrated Halley very accurately revised and corrected both the text of Apollonius, and the translation from the arabic; and he has himself restored the eighth book, conformably to the plan of the author; the whole forming á magnificent edition, which was published at Oxford in 1710.
In the first four books Apollonius treats of the generation of the conic sections, and of their principal properties with respect to their axes, foci, and diameters. Most of these things were already known: but when Apollonius borrows some propositions from his predecessors, he does it like a man of genius, who improves and augments the science. Before him the conic sections were considered only in a right cone: but he takes them in any cone whatever, having a circle for it's base, and he demonstrates several theorems, which are either new, or given in a more general form than they had been before,
The following books contain a number of theorems and remarkable problems, altogether unknown before
his time; and hence Apollonius has merited principally the title of the great geometrician, these I shall quote a few particulars.
In the fifth book Apollonius determines the greatest and the least lines, that can be drawn from a given point to the circumference of a conic section. He at first supposes, that the given point is placed in the axis of the conic section, and he solves a great number of curious problems on this subject, with a simplicity and elegance that cannot be too much admired: he then extends the investigation to cases where the point is situated out of the axis, which affords a new field for problems still more difficult. For instance, in proposition LXII he determines the shortest line that can be drawn from a given point placed within a parabola, and out of the axis, by a very ingenious construction, in which he employs an equilateral hyperbola, which cuts the parabola at the point sought. In the same book we also find the germe of the sublime theory of evolutes, which modern geometry has carried so far.
The subject of the sixth book is the comparison of similar or dissimilar conic sections or portions of conic sections. Apollonius here teaches us to cut a given cone in such a manner, that the section shall have given dimensions; also on a cone similar to a given cone he determines a conic section of given dimensions; and every where we find a simplicity, an elegance, and a perspicuity, which afford infinite satisfaction to the admirers of the ancient geometry,
In the seventh book, of which the eighth was a part or a continuation, Apollonius demonstrates, and it
was here these important theorems appeared for the first time, that, in the ellipsis or hyperbola, the sum or difference of the squares of the axes is equal to the sum or difference of the squares of any two conjugate diameters; and that in either of these curves the rectangle of the two axes is equal to the parallelogram formed about any two conjugate diameters. pass over other propositions, very curious, and not less profound.
The age of Archimedes and Apollonius was the most brilliant era of ancient geometry. After these two great men we meet with no other geometrician of the first order in the period we are now considering: yet there are several others, who enriched geometry with interesting theories or discoveries, and have thus merited the esteem and gratitude of posterity.
It appears, that men of great great inventive powers, too much addicted perhaps to abstract and theoretical speculations in geometry, attached too little importance to the applications that might be made of them in practice. This no doubt was the cause, why the first origin of trigonometry, or that branch of geometry by which we find the relations between the sides and angles of a triangle, has fallen into oblivion. Yet trigonometry affords curious problems, which must naturally have excited the rescarches of the early geometricians. For instance, a person might have wished, or even found it necessary, to ascertain the breadth of a large river, without being obliged, or without having it in his power, to measure it directly; or he might have been desirous of knowing the distance between the summits of two mountains
separated by precipices. Now these, and many other problems of a similar kind, are to be resolved by the construction of a triangle, one of the elements of which shall be the quantity sought, and in which three of the six things that constitute it, namely three sides and three angles, are known; with this condition alone, that among the three things known there is one side of the triangle, which is capable of being measured directly, or having it's length determined by some other known distance. Hence we perceive, that the principles of rectilinear trigonometry are very simple. There are circumstances which indicate, that they were not unknown to the egyptians and we are certain, that they were familiar to the greeks. Beside their use in the mensuration of terrestrial distances, they are applicable to several astronomical problems.
From the consideration of plane triangles the geometrician rose to spherical triangles, or those formed by three arcs of great circles of a sphere intersecting each other; a theory particularly useful in astronomy, to which it is in some degree indispensable. It is also somewhat complicated, because, in a space extended according to the three dimensions, we must discover the ratios of the sides and angles of a triangle, the three sides of which are arcs of a circle. Accordingly the rise of spherical trigonometry was slow. We have no reason to imagine, that it had made any progress, or such at least that deserves notice, before the time of Menelaus, who lived about the year 55, and who was both a skilful geometrician and a great astronomer. He wrote a treatise on Chords, which