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In order to abridge the language of geometry particular signs are substituted for the words which most frequently occur ;
and when we are employed upon any number or magnitude without considering its particular value, but merely with a view to indicate its relation to other magnitudes, or the operations to which it is to be subjected, we distinguish it by a letter of the alphabet, which thus becomes an abridged name for this magnitude.
1. + signifies plus, or added to.
The expression A + B indicates the sum which results from the magnitude represented by the letter A being added to that represented by B, or A plus B.
signifies minus. A - B denotes what remains after the magnitude represented by B has been subtracted from that represented by A.
* signifies multiplied by. A x B indicates the product arising from the magnitude represented by A being multiplied by the magnitude represented by B, or A multiplied by B. This product is also sometimes de noted by writing the letters one after the other without any sigo, thus AB signifies the same as A x B.
The expression Ax (B+C - D) represents the product of A by the quantity B+C-D, the magnitudes included within the parenthesis being considered as one quantity.
indicates the quotient arising from the magnitude representB ed by A being divided by that represented by B, or A divided by B.
A=B signifies that the magnitude represented by A is equal to that represented by B, or A equal to B.
A > B signifies that the magnitude represented by A exceeds that represented by B, or A greater than B.
A< B signifies A less than B.
2A, 3A, &c., indicate double, triple, &c., of the magnitude represented by A.
II. When a number is multiplied by itself, the result is the second power, or square, of this number; 5 x 5, or 25, is the second power, or square, of 5.
The second power therefore is the product of two equal factors; each of these factors is the square root of the product; 5 is the square root of 25.
If the second power be multiplied by its root, the result is the third power, or cube ; 5 X 25, or 125, is the third power of 5.
The third power is a product formed by the multiplication of three equal factors; each of these factors is the cube root of this product; 125 is the prodụct of 5 multiplied twice by itself, or 5 * 5 * 5; and 5 is the cube root of 125.
In general A, being an abbreviation of AX A, indicates the second power or square of A.
VĀ indicates the square root of X, or the number, which being multiplied by itself, produces the number represented by A.
A3, being an abbreviation of AX AX A, indicates the third
power or cube of A.
VA indicates the cube root of A, or the number which, being multiplied twice by itself, produces the number A.
The square of a line AB is denoted by AB.
All numbers are not perfect squares or perfect cubes, that is, they have not square roots or cụbe roots which can be exactly expressed; 19, for example, as it is between 16, the square of 4, and 25, the square of 5, has for its root a number comprehended between 4 and 5, but which cannot be exactly assigned.
In like manner 89, which is between 64, the cube of 4, and 125, the cube of 5, has for its cube root a number between 4 and 5, but which cannot be exactly assigned. Algebra furnishes methods for approximating, as nearly as we please, the roots of numbers which are not perfect powers.
III. 1. When two proportions have a common ratio, it is evi. dent that the two other ratios may be put into a proportion, since they are each equal to that which is common. If, for example, we have
E:F::C:D, then we shall have A:B::E:F.
2. When two proportions have the same antecedents, the consequents may be put into a proportion ; for, if we have
A:E::C:F, by changing the place of the means, these proportions will become
B: E::D:F. IV. Other changes, besides the transposition of terms, may be made among proportionals without destroying the equality of the product of the extremes to that of the means.
1. If to the consequent of a ratio we add the antecedent, and compare this sum with the antecedent, this last will be contained once more than it was in the first consequent; the new ratio then will be equal to the primitive ratio increased by unity. If the same operation be performed upon the two ratios of a proportion, there will evidently result from it two new ratios equal to each other, and consequently a new proportion. Let there be, for example, the proportion
4 : 6 :: 12 : 18 we shall have 6+4:4:: 18 + 12:12,
10:4 : ; 30 , 12. 2. If from the consequent of a ratio we subtract the antecedent, and compare the difference with the antecedent, this last will be contained once less than it was in the first consequent; the new ratio will be equal to the primitive ratio diminished by unity. If the same operation be performed upon the two ratios of a proportion, there will result from it two new ratios equal to each other, and consequently a new proportion. From the proportion
4:6::12 : 18, we thus deduce 6
:4:: 18 or
2:4:: 6 : 12.
There being a proportion among any magnitudes whatever designated by the letters
A:B::C:D, we have, by the above changes,
B-A:A::D-C: C. If we change the place of the means in these results, they will become
B + A:D + C :: A:C
B-A:D-C::A: C; but, by the same change, the proportion
A:C::B:D, and, since the ratios A : C, B: D, are equal, we obtain
B+A:D+C::A: C or :: B:D,
B-A:D-C::A: C or ::B:D, a result which may be thus enunciated. In
any proportion whatever, the sum of the two first terms is to the sum of the two last, and the difference of the two first terms is to the difference of the two last, as the first is to the third, or as the second is to the fourth.
Moreover the two ratios A : C, B: D, being common to the two proportions above obtained, it follows that the other ratios of the same proportions are equal, and that consequently
B +A:D+C::B-A:D-C, or, by changing the place of the means,
B+A:B-A:: D+C:D-C; that is, the sum of the two first terms of a proportion is to their dif* ference, as the sum of the two last is to their difference. For example,
6 +4:6-4:: 18 +12:18 - 12,
10: 2 :: 30 : 6. When the proportion
A:B::C:D, is changed into A:C::B:D,
A and B are the antecedents, C and D the consequents; and the proportions
B-A:D-C::A: Cor::B:D, answer to the following enunciation ;
The sum of the antecedents of a proportion is to the sum of the consequents, and the difference of the antecedents is to the difference of the consequents, as one antecedent is to its consequent ;
Whence it follows, that the sum of the antecedents is to their difference, as the sum of the consequents is to their differences If we have a series of equal ratios
A:B::C:D::E:F, by considering only the two first, which form the proportion
A:B::C:D, we obtain by what precedes
A+C:B + D :: A:B; and, since the third ratio E : F, is equal to the first A : B, we have
A + C:B +D::E:F. If we take the sum of the antecedents and that of the consequents in this last proportion, the result will be
A+C+E:B+D+F::E: For :: A:B. By proceeding in the same manner with any number of equal ratios, it will be seen, that the sum of any number whatever of antecedents is to the sum of their consequents, as one antecedent is to its consequent. V. Let there be any two proportions
E:F::G:H, if we multiply them in order, that is, term by term, the products will form a proportion, thus AXE: BXF:: CxG: D x H,
BXF DXH This is evident, since the new ratios
are respecAXE' CX G