THE ELEMENTS OF EUCLID. BOOK II. DEFINITIONS. 1. EVERY rectangle is said to be contained by any two of the straight lines which contain one of the right angles. SCHOLIUM. As already explained in the scholium to the twenty-seventh definition in the former Book, a rectangle is designated as the rectangle under the two lines by which it is contained. 2. In any parallelogram, either of the parallelograms about the diagonal, together with the two complements, is called a gnomon. THEOREM.-If there be two straight lines (A and BC), one of which is divided into any number of parts (BD, DE, EC), the rectangle under the two lines is equal in area to the sum of the rectangles under the undivided line (A) and the several parts of the divided line (BD, DE, EC). CONSTRUCTION. From the point B draw BF perpendicular to BC (a), and make BG equal to A (b); through G draw GH parallel to BC (c), and through D, E, C, draw DK, EL, CII, parallel to BG (c). (a) I. 11. I. 31. A K LH DEMONSTRATION. It is evident that the rectangle BH is equal to the sum of the rectangles BK, DL, EĤ; but the rectangle BH is the rectangle under A and BC, for BG is equal to A (d); and the rectangles BK, DL, EH, are respectively the rectangles under A and BD, A and DE, A and EC, for each of the lines BG, DK, EL, is equal to A (e). Therefore the rectangle under (d) Constr. (e) I. 34 and Constr. A and BC is equal in area to the sum of the rectangles under A and BD, A and DE, and A and EC. A D с + E B SCHOLIUM. It is of considerable importance that the true relationship or connection between Geometry and Analysis be clearly understood before entering upon the second book of the Elements. The subject of Geometry is magnitude; that of Analysis, i. e., Arithmetic and Algebra, is number and quantity. In order, therefore, to a just understanding of the connection between Geometry and Algebra, we must first obtain a clear notion of the connection between the subject of each, that is, between magnitude and number. Let us take a line AB of any given magnitude, and bisect it in C; and let us again bisect AC in D, and CB in E. We thus obtain four equal lines, which are together equal to the given line AB'; and it is evident that, being all equal, any one of those lines taken four times will be equal to the four lines, or to the original line AB; and we have thus two modes of expressing the magnitude of the line AB, namely, either by saying that it is equal to four lines each equal to AD, or by saying that it equals AB. That is, we may either view the line in its entirety, or we may-having first conceived it as being divided into any number of equal parts-view it as the magnitude of one of those parts repeated that number of times. Each of these parts is termed a unit; the process which we follow in order to determine the number of units in a given line is termed measuring that line; and if the unit is found to be contained any exact number of times in the given line, that is to say, if the unit added on to itself any certain definite number of times forms a line neither longer nor shorter than the given line, but exactly coinciding therewith, the unit is termed the measure of the line; and if the same unit is found to measure any other given line, so that being taken any other certain definite number of times, it forms another line exactly equal to the second given line, that unit is said to be the common measure of both the given lines, and those two lines are said to be commensurable. A unit may be arbitrarily determined, that is to say, we may assume a line of any length that we please for the purpose; but having once determined the magnitude that shall constitute the unit, that magnitude must be considered as fixed and unalterable. Thus, in any particular course of investigation, we may assume for the unit a line a yard in length, or a foot in length, or an inch in length; and having done so, we should express the magnitude of any line by the number of lines (each one yard, foot, or inch, as the case might be) which would form a line equal in magnitude to the given line. Here, then, we see the connection between number and magnitude; number may be regarded as the instrument through the medium of which we estimate and express magnitude. Were we not in possession of the common notion or idea of number, we could only express the magnitude of any given line by the actual exhibition of the line, or of another equal to it; but, having that idea, we are enabled to declare its magnitude by comparing it with another standard line (termed a unit), the magnitude of which is already familiar to us; and we thus make known the magnitude of the given line, by stating the result of that comparison, or the number of those units which the line is equal to. We have hitherto assumed that the given line has, in every case, been equal to a certain number of units; but let us now suppose that, having arbitrarily fixed upon a unit, when we apply it to the given line, as AB, by cutting off successive portions AC, CD, DE, &c., each equal to the unit, we at length arrive at a remaining portion, as FB, which is less than the unit; how are we, in such a case, to determine the magnitude of the given line? The most obvious mode C D E would be, to divide one of the units, as EF, by continually bisecting it, until the portion FB of the given line was found to be equal in length to some certain number of the minute equal parts into which we had so divided the unit; and, in every case in which this process could be carried out, the magnitude of the given line would be determined with the same exactness as if it had at once coincided with any given number of units; for we might regard any one of the small parts into which we had divided the original unit as a new unit, and express the length of the given line by stating the number of such lesser units contained in its length. Thus, if the original unit had been a foot, and on applying it to the given line we had found it contained four times, together with a remainder less than a foot; but that, on dividing the unit into twelve equal parts, the remainder was found to be precisely equal to eight of those twelfth-parts, we could, in such case, declare the length of the original line by stating it to be equal to that of 56 units, each a twelfth of a foot, or one inch, in length. It might be regarded as almost self-evident, that this mode of measuring a line could always be adopted; that, in fact, whatever might be the comparative lengths of the unit AC and the remainder FB, by a sufficiently minute subdivision of the former, we could always arrive at some new unit which would be contained in the latter a certain definite number of times; that if, for instance, it was not found to be equal to any definite number of hundredth-parts, it might be of thousandths, or millionths, or even of some much more minute division. But it may be demonstrated that this frequently cannot be done; that, in fact, certain lines have no unit or common measure, however minute, by which they can be both divided without a remainder; and, when such is the case, they are said to be incommensurable. The following lemma is introduced as an instance. LEMMA. If two straight lines (AB and BD) are the side and diagonal of a square (ABCD), they are incommensurable. CONSTRUCTION. From BD cut off DE equal to AB (a); through E draw EF perpendicular to BD (b), and produce it to cut AB in F. Join AE. H G E (a) I. 3. DEMONSTRATION. Because the triangle ADE is isosceles (c), the angles DAE and DEA are equal (d). Then, because DAF and DEF are both right angles (c), they are equal (e). Therefore if the equals DAE and DEA be taken from the equals DAF and DEF, the remaining angles FAE and FEA are equal (f). And because in the triangle AFE, the two angles FAE and FEA are equal, therefore the opposite sides AF and FE are equal (g). But in the triangle BEF the angles FBE and BFE are evidently each equal to half a right angle, therefore the opposite sides BE and FE are equal (g). Complete the square HE, and on its diagonal FB take FG equal to BE or AF (a); wherefore the excess BE of the diagonal BD beyond the side AB is contained twice in that side with a remainder GB; and GB being itself the excess of the diagonal BF beyond the side HB (of the square HE) is contained twice in that side with a remainder LB, which will again be contained twice in the side KB with a remainder; and this process of subdivision might be carried on with I. Ax. 11. (f) I. Ax. 3. (g) I. 6. out limit, whence it is evident that no common measure can be found for the side and diagonal of a square, but that they are incommensurable. Hitherto we have only mentioned one species of magnitude as being measured by number, namely lines; but every kind of magnitude, whether a surface, solid, or angle, may be so estimated, it being only necessary that the unit arbitrarily fixed upon should be of the same species or kind as the magnitude to be measured; that is to say, the unit for measuring surfaces must itself be a surface; for measuring solids, a solid; for angles, an angle. In these, as in the case of lines, the magnitude of the unit is entirely arbitrary; but it is convenient and usual to take, as the unit for measuring surfaces, a square, the length of whose side is equal to the linear unit employed for the measurement of lines; and for estimating the content of a solid, to employ a cube, any one of whose bounding-planes is equal to the square unit. Thus if a linear foot had been assumed as the measure of a line, the square foot would be employed as that of a surface, and the cubic foot as that of a solid. Let ABCD be a rectangle whose two sides are commensurable, the side AB containing four units, and BC seven units; and let those sides be divided respectively into four and seven equal parts, and lines drawn through the points of division perpendicular to the divided side. Now it is evident that each of the portions into which the rectangle is B thus divided is a square unit, and that as the area or magnitude of a rectangle is expressed by the number of units it contains, that of the rectangle ABCD will be found by estimating the number of squares into which it has been divided. We first perceive that the horizontal lines divide it into four equal portions, and next that each of these portions is again divided into seven equal parts by the vertical lines, so that the total number of subdivisions will be found by taking 7 four times, or multiplying seven by four; that is to say, that the number of square units in any rectangle will be found by multiplying together the two numbers which express in units the magnitude of its two contiguous sides. When the rectangle is a square, the number of units in each of the two contiguous sides being the same, we have to multiply that number by itself to obtain its magnitude in square units; and hence the term square, which is applied in Algebra and Arithmetic to the product of a number multiplied by itself. It is, however, important that the term square in Geometry should not be confounded with that in Arithmetic, for which end we speak of a square in Geometry as the square on a line, that in Arithmetic as the square of a number. It would, however, be better in the latter case to avoid altogether the use of the word square, and to substitute the expression the power of a number. Thus if a represent the number of linear units contained in one side of a square, the product of a multiplied by itself, or a2, will equal the number of square units in the square; and similarly, if b represent the number of linear units contained in one side of a rectangle, and c the number in the contiguous one, b multiplied by c, or b. c, will express the number of square units in the rectangle. It will thus be seen that, by means of this symbolism, we can represent and express the magnitude of any rectangle, and deduce algebraically all the properties investigated geometrically by Euclid. But when we attempt to substitute for b and c their numerical values, it may be found that the two magnitudes which they represent are incommensurable, in which case, as no common measure or unit can exist by means of which their lengths can be stated, no definite numerical value can be given to b and c, and therefore the magnitude of the rectangle cannot be exactly found arithmetically, although, in practice, by a minute subdivision of its sides, its magni tude may be determined within any amount of exactness that may be desired. In considering the properties of lines algebraically, it is necessary that they should always be measured in the same direction, that is, either always from right to left, or from left to right; but if, in the same investigation, it is found necessary to measure one or more of the lines in a contrary direction to the others, the magnitude of such line or lines must be considered as having a negative value, and any algebraical quantity assumed as representing that value must have the minus sign (-) prefixed to it. Having thus pointed out both the connection and the difference between number and magnitude, we shall append to such of the propositions in the second book as admit of being so demonstrated an algebraical investigation and proof. THEOREM. If there be two numbers (a and b), one of which is divided into any number of parts (m, n, p), the product of the two numbers is equal to the sum of the products of the undivided number (a) and the several parts of the divided number (m, n, p). DEMONSTRATION. By the hypothesis b = m + n + p. Multiply these equal quantities by a, then b. a = (m + n + p). a, or ab=am+an+ap. Ꮮ M N P D H 0 Q E I COROLLARY. If two straight lines (AB and BC) be each of them divided into any number of parts (AD, DE, EB, and BF, FG, GC), the rectangle under the two lines is equal in area to the sum of all the rectangles under all the parts of the one, taken separately with all the parts of the other. For the rectangle under AB and BF is equal in area to the sum of the rectangles under BF and AD, BF and DE, and BF and EB; also the rectangle under AB and FG is equal in area to the sum of the rectangles under FG and AD, FG and DE, and FG and EB; and the rectangle under AB and GC is equal in area to the sum of the rectangles under GC and AD, GC and DE, and GC and EB. But the rectangle under AB and BC is equal in area to the sum of the rectangles under AB and BF, AB and FG, and AB and GC; therefore it is equal in area to the sum of all the rectangles under all the parts of the one, taken separately with all the parts of the other. B G C The above corollary, stated algebraically, will be as follows:If two numbers (a and b) be each of them divided into any number of parts (m, n, p, and q, r, s), the product of the two numbers is equal to the sum of the products of all the parts of the one, multiplied by all the parts of the other. Then if we multiply the first member of the first equation by b, and the second member by its equal (q+r+s), we have a. b = (m + n +.p) · (q + r + s), or a b = m q + n g + p q + m r + n r + pr+ms+ns + ps. |