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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; jo part of AB, however small, will measure the segments AC, BC.
For (V. 26.) take AC out of AB, and again the re- A FE 32 9. T; 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 subordinate 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 e- qual to EBF, and the side EF B (I, 11.) equal to BF; but EF is - a Bož 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 D 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.
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.
1. Straight lines drawn from the same point, are termed diverging lines.
2. Straight lines are divided similarly, when their corresponding segments have the same ratio.
3. A straight line is cut in extreme and mean ratio, when the one segment is a mean proportional between the other segment and the whole line.
4. A straight line is said to be cut harmonically, if it consist of three segments, such that the whole line is to one extreme, as the other extreme to the middle part.
5. The area of a figure, is the quantity of space which its surface occupies.
6. Similar figures are such as have their angles respectively equal, and the containing sides proportional.
7. If two sides of a rectilineal figure be the extremes of an analogy, of which the means are two corresponding sides in another rectilineal figure; those figures are said to have their sides reciprocally proportional.