64 EXAMPLES. Ans.. Ans. 10% or 100/10. 20 1. Multiply 6 by 6, and the product is 6%, or V6. 2. Divide 6 by 63, and the quotient is 625 or 3 3 3. Add 32 and 108, and multiply the same 3 20 6. Ans. 16 or 22. 3 4 125 Ans. 52. 4. Add 32 and 108, and divide the sum by 5. Find the shortest method of dividing 3 by 2, to any given place of decimals. 3×√2 3√2 18 4.242640 &c. 1 3 6. Find the sum of ✔ and ✔, and also their difference. Ans. Their sum is 54, or 32. Their diff. is 1√2, or ✓¦. 7. What is the sum and difference of and 32 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 2 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 315; 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 . 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 coup. lets, 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. &, 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: 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 C 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 between 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 GeometriDd 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, 10×15-12-15X12-16, or 10X15-10X12-15X12—15× 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 31. As 10X15-10×12=15×12-15 x 10, then 10×15-15X12+ 15×10=10x 12, or 2X10X15-12X15-10X12, or, 2×10-12×15=10×12, and 15= 10×12 that is, 15, the third is equal to the product of the first and second 2X 0-12' terms, divided by the difference of twice the first term and the second term. 2. Given the 1st and 3d to find the second terri. From the same equivalent expression, we get 2X10X15 154-10X12, and 2X10X15 15×12+10×12 ,12, that is, the second term is equal to twice theproduct 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): 10×15-10×12=15 × 12, or 2× 15—12 15X12 that is, the first term is equal to the product X10=15x12, and 10 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 oclave, 25, the second term. that chord, which is called a fifth. -33-33, the second term, and is the length of If there be four harmonical proportionals, as, 5, 6, 8 and 10; then, according to the definition, 5: 10:6-5: 10-3, and as before, 5× 10-8-10×6-5, or 5X 10-5X8=10×6-10×5. 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; 2X10X5-10×6=5x8, 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, 3, and 10, then E5 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. the terms be as before, 6×10−8x 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 term. 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 couplets, 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 geometrical proportion; for=2=common ratio of the coup2 8 16 2 lets, and 8 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.t 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 in 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—3—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 corresponding terms must be equal to that of the two extremes. THEOREM III. In any continued Arithmetical Proportion, as 4, 4+2, 4+4, 4+6, 4+3, &c. the last or greatest term is equal to the sum of the first or least term and the common difference of the terms, multiplied by the number of the terms less one.* THEOREM IV. The sum of any rank, or series of quantities in continued Arithmetical Proportion (1. 3. 5. 7. 9. 11.) is equal to the sum of the two extremes multiplied into half the number of terms.† ARITHMETICAL PROGRESSION. ANY rank of numbers, more than two, increasing by a common excess, or decreasing by a common difference, is said to be in Arithmetical Progression. If the succeeding terms of a progression exceed each other, it is called an ascending series or progression; if the contrary, a descending series. So And U. 2. 4. 6. 8. 10, &c. is an ascending arithmetical series.. 1. 2. 4. 8. 16. 32, &c. is an ascending geometrical series. 10. 8. 6. 4. 2. 0, &c. is a descending arithmetical series. 32. 16. &. 4. 2. 1, &c. is a descending geometrical series. * For since each term, after the first, exceeds that preceding it by the common difference, it is plain that the last must exceed the first by so many times the common difference as there are terms after the first; and therefore must be equal to the first, and the common difference repeated that number of times. For, because (by the second Theorem) the sum of the two extremes, and that of every other two terms, equally remote from them are equal, the whole series, consisting of half so many such equal sums as there are terms, will therefore be equal to the sum of the two extremes, repeated half as many times as there are terms. The same thing also holds, when the number of terms is odd, as in the series 4, 8, 12, 16, 20; for then, the mean, or middle term, being equal to half the sum of any two terms, equally distant from it on contrary sides, it is obvious that the value of the whole series is the same as if every term thereof were equal to the mean, and therefore is equal to the mean (or half the sum of the two extremes) multiplied by the whole number of terms; or to the sum of the extremes multiplied by half the number of terms. The sum of any number of terms of the arithmetical series of odd numbers 1, 3, 5, 7, 9, &c. is equal to the square of that number. For, 0-1 or the sum of 1 term = 12 or 1 16+9 or the sum of 5 terms = 52 or 25, &c. By continuing the addition, the rule would be true for any number of terms. EXAMPLE. The first term, the ratio, and number of terms given, to find the sum of the series. A gentleman travelled 29 days, the first day he went but 1 mile, and increased every day's travel 2 miles; How far did he travel? 29X29-341 miles, Ans. |