terms, they should be reduced to this form. Thus 4-4=0; 5 5 5 and 3√3—2√3=1√3 or √3. Also 75--45-35. What is the difference between 1350 and 162? Ans. 22. Note 1. Surds, apparently incapable of addition or subtraction except by their signs, may sometimes be reduced to a common surd, by the following process, and their sum and difference readily found. Thus let the surds be ✓ and √ As √√x= √√, and as √3=√={√, then √3+√} = 2√} + {√}; {{}& = {X}√6=√6, their sum: And 2 V6, their difference. What is the sum and difference of √ and ✓? Ans. Their sum is 16. Their diff. is 3√6. What is the sum of 15 and √? Ans. Note 2. If the same quantity is under different radical signs, or if the same radical sign has different quantities under it when the surds are in their simplest terms, the surds can be added or subtracted only by the signs of addition or subtraction. Thus it is 3 evident, that √2+2, is neither twice the square root of 2 nor 3 twice the cube root of 2; and that 3√3—2√3, is neither the square root of 3 nor the cube root of 3. It is equally obvious, that 2 √ 3 +2√2, is neither four times the square of 3 nor of 2; and that 4√2 -2√3, is neither twice the square of 2 nor of 3. 5. Surds of the same radical sign are multiplied like other numbers, but the product must be placed under the same radical sign. Thuş 27×64=27×64=1728=12, for 27=3, and 64 3 3 3 3 3 3 3 =4, and 27x64=3x4=12. And √2 × √3=√2x3 = √6. Also 3√3x4√5=12√15 or √27×√80, and √27X✓80=√27X80 =12√15=√2160. Sometimes the product of the surds becomes a complete power of that root, and the root should then be extracted, as in the first of the preceding examples. Also in this example; √2x√200== √400=20. It is evident from the first example in this section, that, when quantities are under the same radical sign, the root of the product of quantities is equal to the product of their roots. 3 3 3 1 If a surd be raised to a power denoted by the index of the root, the power will be rational. Thus, √3X√3, or 3x3=3. In this example 2 is the index of the root, and the surd is raised to the second power or square. Also 4x4X4 4. If fractional indices be used, in order to multiply surds of the same root, you have only to add the indices. Thus 4x4a×43=4a=41 or 4, unity being the implied index of 4, or of the first power of any number. In all cases when the sum of the numerators contains the com 1 3 mon denominator a certain number of times exactly, the product will As 54 may be expressed according to the notation of powers, 3 44-3 8 thus, 51, and 53 by 51, hence 5x555î=5®. Therefore, to multiply different powers of the same root, you have only to add the indices of the given root, and place the sum for the index of the power which is produced. Thus 32X32=34, or the square of a number multiplied by itself produces the fourth power; the cube by the cube, the sixth power, and so on. 3 5 Thus also 24×2= 6. Surds of the same radical sign are divided like whole numbers, but the quotient must be placed under the same radical sign. Thus 1728-64=√1728-64=√273; and 6÷/3= /6-3=√2. Sometimes the quotient becomes a complete power, as in the first example, in which case the root should be extracted. So also in the following; 100/100/400-100√4=2. 3 3 As 1728=12, and 3 3 641, then 1723/64=12÷4=3= 27. Hence, the quotient of the roots of quantities is the same as the root of their quotient, if the quantities are under the same radical sign. Divide 108 by 6. Now √ 108÷÷√6=√√/ 108÷÷÷6=√18= √9×2=34/2. Divide 100 by 32. Now 9/100=8100, and 3/2= 18, and 8100÷÷÷√18=√3100÷18=√450=15√2. Or 9√100 +39-3X √100√293 × 100+2=93 X √50= 3√50=8×5√2==15√2. 3 3 3 18 3 3 Divide by Now!÷1=?; and √ √ ] = √ M =3V;=3√2; and 3×3√2=3√2. Divide 48 by V; 5√60 by 3√15; and √ by √3. If the quantities under the radical sign be the same, the quotient will be found by dividing the rational parts only. Thus √2÷÷√2 =√1=1, or 2 is contained in 2 once. Also √3÷1√3=2, and 2√5÷5√5=3. To divide one power by another of the same root, place the dif ference of the indices for the index of the given rool. This is merely reversing a process given in the preceding section. The reason of the process may also be seen in the following manner. Thus ing the indices to a common denominator; and 24÷21_√20 __ 24-9 $23 If the index of the divisor exceed that of the dividend, the index of the quotient will be their difference with the sign of subtraction before it. Thus, 52-5552-5-5-3. Now, as 52-55= 55 52 1 52X53 539 1 52 Hence a power whose index has the sign of subtraction before it, is the same power of the reciprocal of that quantity. Hence, there is an obvious method of transferring powers from the numerator to the denominator of a fraction. Thus, method of finding the value of a quantity whose index has the sign of subtraction before it. Thus 2-3: 1 1 2 23 and = 16, 7. To raise surds to any power, multiply the index of the surd by the index of the required power. Thus 23 raised to the square 21+1=2=2×2= =2; and the cube of 33+1+34 If there be rational parts with the surd, they must be raised to the given power, and prefixed to the required power of the surd. Thus, 33, or 3×3 raised to the square, is 3x3+=32×3 =32×3×2=9√32, 2 9. And the cube of 2=21+1+1_ 2a×3=2*=√23=✅✔✅8=2✔✅2, when reduced to its simplest terms. Also, the fourth power of 12 is ×22 = 3 Required the fifth power of ✔}. 8. To extract any root of a surd quantity, divide the index of the quantity by the index of the required root. Thus, the square root of 24 is 22-23 or 2, and the cube root of 34 If there be rational parts with the surd, their root must be prefixed to the required root of the irrational part. Thus, the square root of 99, or 9x91-932-33. The process must evidently be the reverse of that in the preceding section, and the reason of it is obvious. EXAMPLES. Ans.. Ans. 102 or 100/10. 20 1. Multiply by 6, and the product is 6%, or 69. 2. Divide 6 by 6, and the quotient is c or 6. 3 3 20 3 3. Add 32 and 108, and multiply the same by ✔ 3 3 3 Ans 16 or 22. 3 4. Add 32 and 108, and divide the sum by ✔ Ans. 52. 5. Find the shortest method of dividing 3 by 2, to any given place of decimals. 3 3X√2 3√2 √18 4.242640 &c 6. Find the sum of ✔ and ✔, and also their difference. Ans. Their sum is 54, or 3√2. Their diff. is ↓√2, or ✓¦ 3 7. What is the sum and difference of and 3 12 Ans. Their sum is 18. Their diff. 18. 8. There are four spheres each 4 inches in diameter, lying so as to touch each other in the form of a square, and on the middle of this square is put a fifth ball of the same diameter; what is the perpendicular distance between the two horizontal planes which pass through the centres of the balls? Ans. 4 4√2 12 2 =2√2=√8-2-8284+ inches: Note. It may be seen from this example that the diameter of the ball divided by √2, will give the distance between the planes, whatever be the diameter of the ball, or, which is the same, half the diameter of the ball multiplied by the square root of 2. 9. There are two balls, each four inches in diameter, which touch each other, and another, of the same diameter is so placed between them that their centres are in the same vertical plane; what is the distance between the horizontal planes which pass through their centres ? Note. It is evident from this example, that in all similar cases, half the diameter of the ball multiplied by the square root of 3, gives the distance between the planes. 10. There is a quantity to whose square is to be added; of the sum the square root is to be taken and raised to the cube; to this power are to be added, and the sum will be 15; what is that quantity? Ans. ✔. OF PROPORTION IN GENERAL. NUMBERS are compared together to discover the relations they have to each other. There must be two numbers to form a comparison: the number, which is compared, being written first, is called the antecedent; and that, to which it is compared, the consequent. Numbers are compared with each other two different ways: The one comparison considers the difference of the two numbers, and is called arithmetical relation, the difference being sometimes named the arithmetical ratio; and the other considers their quotient, which is termed geometrical relation, and the quotient, the geometrical ratio. Thus, of the numbers 12 and 4, the difference or arith metical ratio is 12--4-8; and the geometrical ratio is of 2 to 3 is 3. 12 - 4 =3, and If two, or more, couplets of numbers have equal ratios, or differences, the equality is termed proportion; and their terms, similarly posited, that is, either all the greater, or all the less taken as antecedents, and the rest as consequents, are called proportionals. So the two couplets 2. 4, and 6, 8, taken thus, 2, 4, 6, 8, or thus, 4, 2, 8, 6, are arithmetical proportionals; and the two couplets, 2, 4, and 8, 16, taken thus 2, 4, 8, 16, or thus, 4, 2, 16, 8, are geometrical proportionals.* To denote numbers as being geometrically proportional, the couplets are separated by a double colon, and a colon is written between the terms of each couplet; we may, also, denote arithmetical proportionals by separating the couplets by a double colon, and writing a colon turned horizontally between the terms of each couplet. So the above arithmeticals may be written thus, 2.. 4 :: 6 .. 8, and 4.. 2: 8.. 6; where the first antecedent is less or greater than its consequent by just so much as the second antecedent is less or greater than its consequent: And the geometricals thus, 2: 4 :: 3 : 16, and 4:2:: 16:8; where the first antecedent is contained in, or contains its consequent, just so often, as the second is contained in, or contains its consequent. Four numbers are said to be reciprocally or inversely proportional, when the fourth is less than the second, by as many times, as the third is greater than the first, or when the first is to the third, as the fourth to the second, and vice versa. Thus 2, 9, 6 and 3, are reciprocal proportionals. Note. It is common to read the geometricals 2: 4 :: 8: 16, thus, 2 is to 4 as 8 to 16, or, As 2 to 4 so is 8 to 16. Harmonical proportion is that, which is between those numbers which assign the lengths of musical intervals, or the lengths of strings sounding musical notes; and of three numbers it is, when the first is to the third, as the difference between the first and second is to the difference between the second and third, as the numbers 3, 4, 6. Thus, if the lengths of strings be as these numbers, they will sound an octave 3 to 6, a fifth 2 to 3, and a fourth 3 to 4. Again, between 4 numbers, when the first is to the fourth, as the difference be tween the first and second is to the difference between the third and fourth, as in the numbers 5, 6, 8, 10; for strings of such lengths will sound an octave 5 to 10; a sixth greater, 6 to 10; a third greater 8 to 10; a third less 5 to 6; a sixth less 5 to 8; and a fourth 6 to 8. Let 10, 12, and 15, be three numbers in harmonical proportion, then by the preceding definition. 10: 15: 12-10: 15-12, and by Theorem I. of Geometri Dd |