ing the indices to a common denominator; and 24÷24: 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 55=53—5—5—3, 52 = 539 53 52X53 Now, as 52-55 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, 4X32, and of subtraction before it. Thus 2-3 method of finding the value of a quantity whose index has the sign 7. To raise surds to any power, multiply the index of the surd by the index of the required power. Thus 2 raised to the square is 21+1=21=212-2; and the cube of 3=3+1+1=34×3 =34. 3 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 3×33 raised to the square, is 32×31+3=32×35 =32X33 =9√32, or 9√9. And the cube of 21=21+1+¦. 23×3=22=√✓23=✅✔✅8=2√2, when reduced to its simplest terms. Also, the fourth power of √2 is 1×22=¿· 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 23 is 222 or 2, and the cube root of 34 3 _33×4—3a or $3. 6 3473 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 9×9932-33. The process must evidently be the reverse of that in the preceding section, and the reason of it is obvious. What is the cube root of 8? Ans. t 20 2. Divide 6 by 6, and the quotient is 6% or 6. 3 3 3. Add 32 and 108, and multiply the same by 3 by✓113. Ans 163 or 22. 4. Add 32 and 103, and divide the sum by✔ Ans. 52. 5. Find the shortest method of dividing 3 by 2, to any given place of decimals. Now 3 3X√2 3/2 18 4.242640 &c 72√2×2 2 &c. Ans. Their sum is √54, or √2. Their diff is√2, or 3 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. 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 arith12 metical ratio is 12-4-8; and the geometrical ratio is of 2 to 3 is 3. 4 3, and If two, or more, couplets of numbers have equal ratios, or differences, the equality is termed proportion; and their terms, simiJarly 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.. 28.. 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: 48: 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 S 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: 15-19, and by Theorem 1. of GeometriD d Proportion is distinguished into continued and discontinued. If, of several couplets of proportionals, written down in a series, the difference or ratio of each consequent, and the antecedent of the next following couplet, be the same as the common difference or ratio of the couplets, the proportion is said to be continued, and the numbers themselves, a series of continued arithmetical or ge cal Proportion, 10×15—12=15×iz-10, or 10×15—10×12=15×12-15X 10, whence if any two of the three terms be given, the other may be found in the following manner. CASE 1. Given the 1st and 2d terms to find the 3d. As 10×15-10×12=15×12-15 x 10, then 10×15-15 × 12+ 15×10=10x 12, or 2×10×15-12X 15-10x12, or, 2X10-12X15=10X12, and 1510X12 that is, 15, the third is equal to the product of the first and second 2X10-12 terms, divided by the difference of twice the first term and the second term. 2. Given the 1st and 3d to find the second term. From the same equivalent expression, we get 2×10×15=15×12+10×12= 2X10X15 154-10X12, and =12, that is, the second term is equal to twice the 104-15 product of the first and third terms, divided by the sum of the first and second terms. 3. Given the second and third to find the first term. From the same expression, we get 2×10X15-10 × 12—15×12, or 2× 15—12 15X12 X10=15×12, and 10 =, that is, the first term is equal to the product 2X15-12 of the second and third terms, divided by the difference of twice the third term and the second term. Ex. Find from third term, or monochord, 50, and the first term, or octare, 25, the second term. 2500 33-33, the second term, and is the length of that chord, which is called a fifth. If there be four harmonical proportionals, as, 5, 6, 3 and 10; then, according to the definition, 5: 10 :: 6-5 : 10-3, and as before, 5X 10-8-10×6-5, or 5X 10-5×8=10x6-10×5. From this expression, we may find any one of four harmonical proportionals from the other three. Thus, the first three being giv5X8 en to find the fourth; 2×10×5-10×6=5×8, and 10=2X5-6 , that is, the fourth term is equal to the product of the first and third divided by the difference of twice the first term and the second term. In the same manner, it may be shown, that the third term of four harmonical proportionals is equal to the difference of twice the product of the first and fourth terms and the product of the second and fourth terms, divided by the first term. 2X5X10-6 × 10 If the terms be 5, 6, 8, and 10, then 85 Also, The second term is equal to the difference of twice the fourth and the third term, multiplied by the quotient of the first divided by the fourth term. It the terms be as before, 6=2× 10—8×-- 5 Also, The first term is equal to the product of the second and fourth terms. divided by the difference of twice the fourth and the third terin. Thus 5 6X10 2X10-8 ometrical proportionals. So 2, 4, 6, 8, form an arithmetical progression; for 4-2-6-4-8--6=2; and 2, 4, 8, 16, a geometrical progression; for ===2. But, if the difference or ratio of the consequent of one couplet, and the antecedent of the next couplet be not the same as the common difference or ratio of the complets, the proportion is said to be discontinued. So 4, 2, 8, 6, are in discontinued arithmetical proportion; for 4-2=8-6=2=common difference of the couplets, 8-2-6=difference of the consequent of one couplet and the antecedent of the next; also, 4, 2, 16, 8, are in discontinued 4 16 =2=common ratio of the coup geometrical proportion; for 28 lets, and 16 = ratio of the consequent of one couplet and the antecedent of the next. ARITHMETICAL PROPORTION. THEOREM I. IF any four quantities 2, 4, 6, 8 be in arithmetical proportion,* the sum of the two means is equal to the sum of the two extremes.† And if any three quantities, 2, 4, 6, be in arithmetical proportion, the double of the mean is equal to the sum of the extremes. THEOREM II. In any continued Arithmetical Proportion (1, 3, 5, 7, 9, 11) the sum of the two extremes, and that of every other two terms, equally distant from them, are equal. Thus, 1+11=3+9=5+7+ When the number of terms is odd, as in the proportion 3. 8. 13. 18. 23, then, the sum of the two extremes being double to the mean or middle term, the sum of any other two terms, equally remote from the extremes, must likewise be double to the mean. * Although in the comparison of quantities according to their differences, the term proportion is used: yet the word progression, is frequently substituted i its room, and is indeed more proper; the former form being, in the common acceptation of it, synonymous with ratio, which is only used in the other kind of comparison. + For since 4-2-8-6, therefore 4+6=2+8. Since, by the nature of progressionals, the second term exceeds the first by just so much as its corresponding term, the last but one, wants of the last, it is evident that when these corresponding terms are added, the excess of the one will make good the defect of the other, and so their sum be exactly the same with that of the two extremes, and in the same manner it will appear that the sum of any two other correspon-ling terms must be equal to that of the two extremes. |