sure, or they are incommensurable. But, as the residue of the subdivision is necessarily diminished at each step of this operation, it is evident that some element may always be discovered, which will measure A and B nearer than any assignable limit. PROP. XXVII. PROB. To express by numbers, either exactly or approximately, the ratio of two given homogeneous quantities. Let A and B be two quantities of the same kind, whose numerical ratio it is required to discover. Find, by the last proposition, the greatest common measure E of the two quantities; and let A contain this measure K times, and B contain it L times: Then will the ratio K : L express the ratio of A : B. For the numbers K and L severally consist of as many units, as the quantities A and B contain their measure E. It is also manifest, since E is the greatest possible divisor, that K and L are the smallest numbers capable of expressing the ratio of A to B. If A and B be incommensurable quantities, their decomposition is capable at least of being pushed to an unlimited extent; and, consequently, a divisor can always be found so extremely minute, as to measure them both to any degree of precision. PROP. XXVIII. THEOR. A straight line is incommensurable with its segments formed by medial section. If the straight line AB be cut in C, such that the rectangle AB, BC is equivalent to the square of AC; no part of AB, however small, will measure the segments AC, BC. For (V. 26.) take AC out of AB, and again the re- A. J. E. P_{! I. mainder BC out of AC. But -AD, being made equal to BC, the straight line AC is likewise divided in D, by medial section (II. 19. cor. 1.); and, for the same reason, taking away the successive remainders CD, or AE, from AD, and DE or AF from AE, the sub ordinate lines AD and AE are also divided medially in the points E and F. This operation produces, therefore, a series of decreasing lines, all of them divided by medial section: Nor can such a process of decomposition ever terminate; for though the remainders BC, CD, DE, and EF continually diminish, they must still constitute the segments of a similar division. Consequently there exists no final quantity capable of measuring both AB and AC. Cor. Since (V. 6. and V. 24.) the whole line is to its smaller segment in the duplicate ratio of the same line to its greater segment, it evidently follows that the squares of the parts of a line divided by medial section are likewise mutually incommensurable. PROP. xxix. THEOR. The side of a square is incommensurable with its diagonal. Let ABCD be a square and AC its diagonal; AC and AB are incommensurable. For make CE equal to AB or BC, draw (I. 5. cor.) the perpendicular EF, and join BE. Because CE is equal to BC, the angle CEB (I. 10.) is equal to CBE; and since CEF and CBF are right angles, the remaining angle BEF is equal to EBF, and the side EF B (I. 11.) equal to BF; but EF is also equal to AE, for the angles EAF and EFA of the triangle AEF are evidently each of them half a right angle. Whence, making FH equal to FB, FE or AE,-the excess AE of the diagonal AC above the side AB, is contained twice in AB, with a remainder AH; and AH again, being the excess of the diagonal AF of the derived or secondary square GE above the side AE, must, for the same reason, be contained twice in AG, with a new remainder AL; and this remainder will likewise be contained twice with a corresponding remainder in AH, the side of the ternary square KH. This process of subdivision is, therefore, interminable, and the same relations are continually reproduced. ELEMENTS OF GEOMETRY. BOOK WI. The doctrine of Proportion, grounded on the simplest theory of numbers, furnishes a most powerful instrument, for abridging and extending mathematical investigations. It easily unfolds the primary relations subsisting among figures, and those of the sections of lines and circles; but it also discloses with admirable felicity that vast concatenation of general properties, not less important than remote, which, without such aid, might for ever have escaped the penetration of the geometer. The application of Arithmetic to Geometry forms, therefore, one of those grand epochs which occur, in the lapse of ages, to mark and accelerate the progress of scientific discovery. |