S 1 N A : 



demonBtrationa, a continual and easy reference to the figure. 

 .itfui-.li tli- i aw generally mod for 



\M poculiur tn t-Jcuoietry ; and tmall lettert for that 

 which i- j>n>|*>rl> ulp-l. 



If a line, whoso length u measured from a given )x>int or lino, 

 bo 1 positive, a lino i>r- 



oeoding in the opposite direction 

 must be con itive. It 



A B (Fig. 2), rookonod from DM on 

 the ri-jht, ia positive, AC on the 

 I, ft ia negative. Hence, if in the 

 B course of a calculation the alge- 

 braical value of a lino is found to 

 bo n<v/a/iiv, it mast be measured 

 in a direction opposite to that 

 which, in the same process, has 

 been considered positive. 

 P i( ^ o. I" algebraical calculations there 



is frequent occasion for multij)li- 



cation, division, involution, etc. But how, it may bo asked, 

 win ijt-fif'tfical quantities be multiplied into each other? 

 One of the factors in multiplication is always to be con- 

 sidered as a number. The operation consists in repeating the 

 multiplicand as many times as there are units in the multi- 

 plier. How, then, can a line, a surface, or a solid, become a 

 multiplier ? 



To explain this it will be necessary to observe, that whenever 

 one geometrical quantity is multiplied into another, some par- 

 ticular length is to be considered the unit. It is immaterial 

 what this length is, provided it remains the same in different 

 parts of the same calculation. It may be an inch, a foot, a rod, 

 or a mile. If, for instance, one of the lines bo a foot long, and 

 the other half a foot, the factors will be, one 12 inches, and 

 the other 6, and the product will bo 72 inches. Though it 

 would bo absurd to say that one lino is to be repeated as 

 often as another is long, yet there is no impropriety in saying 

 that ono is to be repeated as many times as there are feet 

 or rods in the other. This tho nature of a calculation often 

 requires. 



If the line which is to be the multiplier is only a part of the 

 length taken for the unit, the product iz a like part of the 

 multiplicand. Thus, if one of the factors is 6 inches, and the 

 other half an inch, the product is 3 inches. 



Instead of referring to tho measures in common use, as 

 inches, feet, etc., it is often convenient to fix upon one of the 

 lines in a figure as the unit with which to compare all the 

 others. When there arc a number of lines drawn within and 

 about a circle, the radius is commonly taken for the unit. This 

 ia particularly the case in trigonometrical calculations. 



The observations which have been made concerning lines 

 may be applied to surfaces and solids. There may be occasion 

 to multiply tho area of a figure by the number of inches in 

 some given line. 



But hero another difficulty presents itself. The product of 



two lines is often spoken of as being equal to a surface ; and 



the product of a line and a surface as equal to a solid. But if 



a line has no breadth, how can the 



D C multiplication that is, the repeti- 



tion of a line produce a surface ? 

 And if a surface has no thickness, 

 how can a repetition of it produce 

 a solid ? 



In answering these inquiries it 

 must be admitted that measures of 

 length do not belong to the same 

 class of magnitudes with superficial 

 or solid measures ; and that none of 

 the stops of a calculation can, pro- 

 perly speaking, transform the one 

 into the other. But though a line cannot become a surface or 

 a solid, yet the several measuring units in common use are so 

 adapted to each other, that squares, cubes, etc., are bounded by 

 lines of the same name. Thus the side of a square inch is a 

 linear inch ; that of a square rod, a linear rod, etc. The length 

 of a linear inch is therefore the same as the length or breadth 

 of a square inch. 



If, then, several square inches are placed together, as from Q 

 to R (Fig. 3), the number of them in the parallelogram o B is tho 



am* M tho number of linear inchos ia the aid* <. E ; and if we 

 know tho length of thu, we have, of coarse, the area of thr 

 parallelogram, which U ban supposed to be one inch wide. 



liut if tho breadth u several inches, the larger paralUlofraa 

 contains as many smaller ones, each an inch wide, M there are 

 inches in the whole breadth. Thu*, if the parallelogram AC 

 (Fig. 3) is 5 inches long and 3 inches broad, it may b* 

 divided into three such parallelograms MOB. To obtain, then. 

 the number of squares in the Urge parallelogram, we have only 

 to multiply the number of squares in one of the small psrallalo 

 grams into the number of snob iiaiallakijiiiiH ""tttained in thr 

 whole figure. But the number of square inches in one of the 

 small parallelograms is equal to the number of linear inches fa 

 the length A B. And the number of small parallelograms is 

 equal to the number of linear inches in the breadth B c. It is 

 therefore said concisely, that the area of a parallelogram it 

 equal to its length multiplied into its breadth. 



We hence obtain a convenient algebraical expression for the 

 area of a right-angled parallelogram. If two of the sides per- 

 pendicular to each other are A B and B c, the expression for the 

 area is A B X B c ; that is, putting a for the area, 



a = A B X B c. 



It muot be remarked, however, that when A B stands for a 

 line, it contains only linear measuring units; but when it 

 enters into the expression for the area, it is supposed to contain 

 superficial units of the same name. 



The expression for the area may also be derived by another 



Fig. 4. 



method more simple, but less satisfactory perhaps to 

 Let a (Fig. 4) represent a square inch, foot, rod, or other 

 measuring unit, and let 6 and I be two of its sides ; also, let A 

 be the area of any right-angled parallelogram, B its breadth, and 

 L its length. Then it is evident that, if the breadth of each 

 were the same, tho areas would be as the lengths ; and if the 

 length of each were the same, the areas would be as the 

 breadths. 



That is, A : a : : L : I, when the breadth is given ; 

 And A : a : : B : b, when the length is given : 



Therefore, A:a::BXL:6xZ, when both vary. 

 That is, the area is as the product of the length and breadth. 



Hence, in solving problems in Geometry, the term product 

 is frequently substituted for rectangle ; and whatever is there 

 proved concerning the equality of certain rectangles, may be 

 applied to the product of the lines which contain the rectangles. 



Tig. 5. 



- 



The area of an oblique parallelogram is also obtained by 

 multiplying the base into the perpendicular height. Thus the 

 expression for the area of the parallelogram A B x x (Fig. 5) ia 

 M N X A D, or A B X B c. For A B X B c ia the area of the right- 

 angled parallelogram A B c D ; and by Euclid I. 36, parallelo- 

 grams upon equal bases, and between the same parallels, arc 

 equal ; that is, A B c D is equal to A B x si. 



The area of a square is obtained by multiplying one of the 

 sides into itself. Thus the expression for the area of the square 

 A c (Fig. 6) is (A B) 3 ; that is, o = (A B) 2 . 



For tho area ia equal to A B X B c. 



But A B = B c ; therefore, A B X B c = A B X A B = (A B)*. 



The area of a triangle ia equal to Tiatfthe product of the base 



