B given in one particular case by Pythagoras, in the problem to which we have already alluded as being that upon which all geometrical measurement depends. It is as follows: We have before us the triangle ABC, of which the sides AB and BC are given, and in which the angle ABC=90%. We have seen that the third line AC is fixed in length. It is now required to ascertain what the length is, in doing which, we shall simply accept and illustrate the answer of Pythagoras. “ The square,” says he,“ upon the hypothenuse of a right angled triangle (that is, upon the side which is opposite the right angle) is equal to the sum of the squares of the other two sides." Which may be written thus : AC’=AB? +BC?. And this of course determines the length of AC, the lengths of AB and BC being given. For example: If AB=40ft., and BC=30ft. Then AC2=402 +30o =2500. AC=50ft. Such is the famous problem to which we have so often made allusion. It is said that the philosopher sacrificed a hecatomb of oxen on discovering it; and well indeed he might! We shall content ourselves with the enunciation of it, which we have already given, without attempting more, as the proof which is found in the 47th Proposition of Euclid's first book involves most of the propositions which precede it, and must be read in its regular course there. It can scarcely, however, be supposed that the order in which it is now found, or indeed that the order in which any of Euclid's Propositions are found is that in which they were first written by their authors. That any man, or any series of men should think out the various problems in the perfectly faultless logical sequence in which they are now arranged, is like imagining that a painter engaged on the portrait of a man should begin by = accurately drawing his boots, or that a child when he is born consists entirely of feet, and gradually developes as he grows up into legs, body, and head. Men think confusedly, grasping a truth here and there, and then reasoning up to it. And there are but few, comparatively speaking, who can analyse their own ideas, even after years of thought, far less think in that analysed logical manner which such a supposition as that made above would imply. And thus it would in truth be of no small interest not only to mathematicians, but even as regards the mind itself, if we could tell the process by which this discovery was made. Whether it was by observation, or by a happy conjecture, as Kepler discovered the laws of planetary motion, or by actual measurement (which would seem most probable) as Galileo found out the area of a parabolic section, in the silence of history we know not. But one thing would seem to us almost incredible ; and that is, that the philosopher found the other forty-six propositions arranged for him in their present order and then deduced this one from them. Far more probable does it seem that this proposition was one of the first to be apprehended and those which are previous in order, to have been merely written in order to prove this principal truth. But we have said enough with regard to the importance of this problem, and now proceed to a brief consideration of its results, And this will form the conclusion of the treatise, as we shall then have seen not only what conditions are sufficient to arrest a moving point, but also by what means the position of the point when arrested may be ascertained. S Ε Ο ΤΙ Ο Ν V. CHAPTER XXXIII. GENERAL SURVEY OF THE RESULTS OF THE PROBLEM OF PYTHAGORAS. The main difficulty of our task is now accomplished, and it only remains to take a short survey of the prospect which this proposition opens out to us. The prospect somewhat resembles, if I may be allowed the comparison, that which greets the traveller who ascends the Mount Brevan from the valley of Chamounix. Immediately upon his arrival at the summit of the long “ ladder" of steps in the face of the rock which forms the last ascent, the whole of the country which he has just left is hidden from his sight, and the great plain of Sixt lies spread out before him. A few steps further, and the valley of Chamounix again bursts on his view, now no longer confined to the part shut in by the neighbouring mountains, but stretching away to where the Alps in the distance tower on either side of the valley of the Rhone. In a somewhat similar manner the student who has mastered this proposition is introduced at once to problems of Trigonometry, while those of Euclid are completely lost to him. He will proceed, however, but a short distance before he finds himself again engaged in geometrical inquiries, now no longer confined to the lines hitherto considered, but embracing in their scope the ranges of Infinitesimal Calculus. Let us give a word at parting to each of these subjects. In the last proposition this problem was solved. Given the position of one point, and the relation of 1. two other points to it : what is the relation of these points to each other? Given A, AB, AC, and CAB, what is the length of BC ? (1.) The answer has been given, when the two lines AC and AB are B in A one particular relation to each other, 2. that namely when the angle between them CAB, is 90° (2.) But from this is easily deduced, as we have already observed, the modification of the result when AB and AC make an angle with each other of any number of degrees. This consideration indeed which forms the chief subject in Euclid's second Book, is hardly to be considered a fresh proposition, but rather an extension of the former. Given therefore the relation of any two points to a third, we can calculate the we D relation of these two between themselves. And since, as have seen, the Area, that is the space which is enclosed by joining any three such points is fixed, 3. or definite, this may be calculated also. If C D then there be a fourth point, the relations of which to two of the others are given (3) it is but necessary to join those between themselves, and considering this А fresh space as a new triangle, to add to it the former and thereby obtain the area of the whole figure, of which the four points are the boundaries (4.) For example: AB and AC are given 4. in length and position, then the space ABC can be determined; DC and DB are given, then the space DCB can be determined, and hence the whole figure ACDB can be determined. А. Nor of course is this method applicable only to four points. It can be used for any number, and thus the size of any polygon can be determined, the number and the length of the sides of which known. Suppose then the number of sides of a regular polygon gradually increases until it becomes infinite. The principle still holds good, the only modification necessary being that which arises from the side becoming too small to be appreciable. And as this is supplied by Infinitesimal Calculus we have the means of determining that well known puzzle,—the area of a circle. Each of these calculations is the subject of trigonometrical reasoning, and cannot therefore be now further pursued. One other trigonometrical application of this proposition is too important to be entirely passed over. But as a full understanding or even statement of it would require a knowledge of a most complicated idea, namely proportion, we shall do no more than B are |