in a given point, and having its circumference bisected by that of the given one. 124. If an angle of a triangle be 60o, the square of the opposite side is less than the squares of the other two by their rectangle : but if an angle be 120°, the square of the opposite side is greater than the squares of the others by their rectangle. 125. In the figure in page 249, prove that the three straight lines joining AF, BE, CD are all equal. 126. The chord of 120° is equal to the tangent of 60°. 127. The sines of the parts into which the vertical angle of a triangle is divided by the straight line bisecting the base, are reciprocally proportional to the adjacent sides. Show from this how a given angle may be divided into two parts, having their sines in a given ratio. 128. The diameter of the circle described about any triangle is equal to the product of any side and the cosecant of the opposite angle. 129. In any triangle ABC, the radius of the inscribed circle is sin Bsin C to sin A sin C equal to a. COSA or, finally, to the cube root of cos B or to c. sinA sin B abc sinA sinB sin C tan A tan B tan C. 130. Given the sum of the tangents, and the ratio of the secants, of two angles to a given radius; to determine the angles geometrically, and by computation. 131. To divide a given angle into two parts having their tangents in a given ratio. 132. Find an angle, such that its tangent is to the tangent of its double, in a given ratio; suppose that of 2 to 5. 3 133. The square of the diameter of a globe is three times the square of the side of the inscribed cube. SKETCH OF THE HISTORY ОР ELEMENTARY GEOMETRY. WITH a few of the facts and relations of Geometry, men must have been acquainted from the earliest period. It is impossible to ascertain, however, in what country the elements of the science first began to be reduced to anything like a systematic form; but, while there is reason to believe, that the ancient Hindoos and Egyptians did something in this way, we have no authentic record regarding anything that was done on the subject, prior to the time at which the study was introduced in ancient Greece, upwards of six hundred years before the commencement of the Christian era. In that country, and in its dependencies, polite literature and the fine arts were cultivated at a very early period with extraordinary zeal and success; and, at the same time, the ingenious and intellectual people of the same regions made great progress in enriching And extending geometry, almost the sole branch of pure science to which the energies of the human mind were directed at an early period in the history of man. Thales of Miletus, in Asia Minor, who is said to have been born about 640 years before Christ, is generally regarded as the first who introduced the study of geometry as a branch of science among the Greeks; and it is probable, that he got his first notions of the subject in Egypt, in which country he is said to have resided for some time, and to have formed an acquaintance with its priests, the sole depositories of whatever science existed in the land. This philosopher, having returned to his native country, Ionia, founded in it what has thence been called the Ionian school. The celebrated Pythagoras, a native of the island of Samos, is said to have been born about 580 years before Christ. This distinguished individual, after having been a pupil of Thales, increased his knowledge by travelling in Egypt and India, and making himself acquainted with the learning which then existed in these countries. To him has been attributed the discovery of the two important propositions, the 32d, and the 47th, of the first book of Euclid. Some have thought, however, that he learned them on his travels.* * Even if Pythagoras did discover these propositions, it is highly improbable, that, as has been related, he sacrificed, in his joy on discovering the 47th, a hundred Another mathematician of considerable celebrity was Hippocrates, of the island of Chios, who lived above 400 years before Christ. He is the first who is known to have succeeded in effecting the quadrature of a curvilineal space, by finding a rectilineal one equal to it. This ingenious, though barren discovery, is given in the 43d proposition of the first book of the Appendix. One of the most distinguished promoters of science among the Greeks was the celebrated philosopher Plato. There is no decisive evidence, that he made mathematical discoveries himself; though some have supposed, that geometrical analysis (see Appendix, book II. prop. 10, &c.) as a separate subject of study, and the conic sections, which were certainly first studied in his school, were indebted to him for their origin. However this may be, he greatly encouraged the study of geometry among his pupils, all of whom he required to have a competent knowledge of the science, and he even put an inscription over the door of his school, stating, that no one unacquainted with geometry should be allowed to enter it as a pupil. The conic sections, as the name, in part at least, implies, are those curves which are produced by the section of a cone by a plane: and the discovery of these important lines is said to have originated in the attempts that were made to solve the celebrated problem, of finding two mean proportionals between two given magnitudes.* oxen to the muses. So expensive a sacrifice, there is good reason to believe, would have been inconsistent with his worldly circumstances: and the shedding of blood was at variance with his principles; as, in consequence of his holding the doctrine of the transmigration of souls, he considered it inhuman to kill creatures which might possibly be animated by the souls of the fathers, brothers, or other relations of the persons who sacrificed them. Thus, Ovid represents him, in the fifteenth book of the Metamorphoses, as saying, Corpora, quæ possint animas habuisse parentum, * Bos aret; aut mortem senioribus imputet annis; Pythagoras paid much attention also to astronomy, music, and arithmetic; and he was certainly one of the most distinguished men of ancient times. * The method of finding one mean proportional between two straight lines is given in the 13th proposition of the sixth book of Euclid. The finding of two mean proportionals, however, by means of elementary geometry, that is, by employing only the straight line and the circle, long exercised the ingenuity of geometricians, but in vain; and all the well-informed mathematicians of modern times agree, that the solution cannot be effected without the aid, directly or indirectly, of the higher geometry. The solution is easily effected by means of the conic sections and many other curves. It is easily obtained, also, by means of algebra. Thus, let a and b be the given magnitudes, and 1 to x the ratio of each of the proportionals to the one following it. Then, as 1:x:: a: ax, the first mean; as, 1: z :: az ax2, the second; and 1:x:: ax2 : ax3; which, by the nature of this problem, is equal to b. Putting it, therefore, equal to b, dividing by a, and extracting the cube root, we find the value of r; and thence the values of the means, ar and az2, are obtained at once. If, for example, it were required to find two mean proportionals between two lines of 54 and 16 inches respectively, we get by dividing the latter by the former, the cube root of which is . Hence, the first mean is 36, two thirds of 54; and the second 24, two thirds of 36. Another celebrated problem which occupied much attention in the school of Plato, and in subsequent times, was the trisection of an angle. The geometricians of that school failed, as all others have done, in solving this problem by means of elementary geometry. While they failed, however, in their main object, both in reference to this problem, and that of finding two mean proportionals, their time and exertions were not thrown away, as they made valuable discoveries regarding the conic sections, as already mentioned, and also in other branches of geometry.* About 333 years before Christ, and 15 years after the death of Plato, the city of Alexandria in Egypt was founded by Alexander the Great. This city was soon after made the royal residence by Ptolemy Lagus, called also Ptolemy Soter, king of Egypt, and that prince established its celebrated Museum. This was an academy in which a society of learned men devoted themselves to the study of science; and it continued to be the asylum of learned men, and the chief seat of mathematics, philosophy, and literature in general, for nearly 1000 years, till the city was taken by the Saracens in the 640th year of the Christian era. Of the distinguished men who were connected with this establishment, the one who claims most attention in this sketch, is Euclid, the author of the Elements, and several other works. Where this mathematician was born, and the times of his birth and death are all unknown. It is believed, however, that he was born upwards of 300 years before the commencement of our era, According to an ancient tradition, the problem of finding two mean proportionals had its origin in a response given by the oracle of Apollo at Delos (whence it was often called the Delian Problem) in reply to an inquiry regarding the means of averting a plague, by which Attica was desolated, in consequence of an affront alleged to have been offered by the Athenians to that deity. In the response, they were ordered to double the altar; and, as the altar of Apollo at Athens was of a cubical form, they were thus required to double a cube. It was soon seen, that if they doubled the length, breadth, and height of the altar, its magnitude would be increased eight fold; and it was discovered, it is said, by Hippocrates of Chios, that if each of these dimensions were increased in the ratio of 1 to the first of two mean proportionals between 1 and 2, the content or volume of the altar would be doubled. * The mode of bisecting an angle is given by Euclid in the ninth proposition of the first book; and that of trisecting a right angle, or any angle obtained by bisecting a right angle, its half, its fourth part, &c. is pointed out in the seventh corollary to the thirty-second proposition of the same book, and in the note to that corollary. The trisection of other angles has been often attempted by means of elementary geometry, hut in vain. By means. however, of the conic sections, as well as of many other curves, the trisection may be readily effected; and the same may be easily done by means of algebra in connexion with trigonometrical principles. To show how the solution may be obtained in the last-mentioned way, let 3A be changed into in the second of each of the sets of formulas marked (235) and (239) in my Elements of Plane and Spherical Trigonometry; and there will be obtained, 4 cos3-3 cos/-cos/=0, and 4 sin3}/—3 sin}/+sin/=0. Then the resolution of the first of these equations, which is a cubic in reference to cos, or of the second, which is also a cubic in reference to sing, will determine 19 by means of its cosine in the one case, and its sine in the other, when is given. Each equation will have three roots, which in the first will be the cosines, and in the second the sine, of one third of 6, one third of +360°, and one third of 0+720°. Other solutions might be obtained from formula (47) or (48) of the same work. and he was the preceptor and friend of Ptolemy Philadelphus, king of Egypt. Of the works which he composed, the most celebrated is the Elements already referred to; a work which has been translated into the languages of all nations that have made any considerable progress in civilization, since it was first published, and which has been more generally used for the purposes of teaching, than any work on abstract science that has ever appeared. What share Euclid had in the discovery of the propositions contained in this work, it is impossible to determine. It is known, that before his time, Hippocrates of Chios, Eudoxus, and others, had composed treatises on elementary geometry, which were highly esteemed, and which must doubtless have contained most of the principal propositions which we find in Euclid. Besides this, the progress made in the theory of the conic sections before his time proves incontrovertibly, that much must have been done in elementary geometry, as its leading propositions are required in establishing that theory. With regard to the composition of the work, however, Euclid is entitled to the greatest credit. His reasoning is conducted with extreme rigour, and the work exhibits throughout such accuracy of thought and expression as is calculated to produce the most beneficial effects on the mind of the careful reader. It is a decisive proof, also, of the estimation in which it has been held, that it has been handed down to modern times, while the similar works that preceded it have been lost; and that even amid the gigantic strides made in modern science, no work has yet appeared which has superseded it in any considerable degree. Of these, This celebrated work is divided into fifteen books. the first four treat of the equality or inequality of straight lines, of angles, and of figures bounded by straight lines or circular arcs. The fifth contains the theory of proportion in the abstract, and the sixth applies this theory in reference to the magnitudes treated of in the first four books. The seventh, eighth, and ninth books treat of the properties of numbers, and form the most ancient tract on arithmetic extant. In the present state of science, however, these books are of scarcely any value. The tenth is devoted to the consideration of commensurable and incommensurable magnitudes, and the last five treat of solids. The last two of these are supposed to have been written, not by Euclid, but by Hypsicles of Alexandria, who lived about 200 years after him. The first six books, and the eleventh and twelfth, form the most valuable part of the work-the part which is found to be of importance in science in its present state; and accordingly most of the modern editions contain only these. The editions of Barrow, Gregory, and Peyrard, besides others, contain the whole fifteen books. Since the time of Euclid many curious and interesting propo A remarkable reply which he is said to have given to Ptolemy, is recorded, and is often referred to. That prince having asked Euclid whether there was no shorter or easier means of acquiring a knowledge of geometry, than that which he had given in his Elements, Euclid is reported to have said in reply," No, Sire, there is no royal road to geometry." |