under the seventh power of the transverse diameter and square of the conjugate a maximum; required the dimensions and area of the ellipse. 178.-A circular fish-pond is to be dag in a garden, that shall take up just half an acre ; what must the length of the chord be that strikes the circle ? 179.-A carpenter is to put an oaken curb to a round well at 8d. per square foot, the breadth of the curb is to be 7 inches, and the diameter within 3] feet; what will be the expence? 180.-Required the quotient of a5 + ms divided by a tm. 181.-What is the quotient of a6 — mo divided by a tm? 182.--Required the quotient of as — ms divided by @- - m. 183.–To find the cube root of the sum of the squares of all the right sines, in a quadrant of a circle whose radius is 40 feet. 184.-Suppose Lewes from Brighton to be 8 miles ; Newbaven from Brighton 9 miles ; Lewes from Newhaven 7 miles: two travellers set out at the same instant of time, one from Newhaven towards Brighton at the rate of 5 miles an hour, the other from Brighton towards Lewes at the rate of 4 miles an hour. How far would each be on his journey when they are the nearest possible to each other? 185 –Given x3 + y 3 : *3 — y3 :: 234 : 109 and xy? = 175 ; ✓ (a? -*) 186.-Simplify, by means of logarithms, the expression (a + x)? 187.- A lady, wealthy, kind, and fair, Your aid, dear sirs, wou'd gladly share, And show what science can achieve. 188.--Cut the greatest cylinder possible from a given parabo oid, and find the centre of gravity of the remainiug solid. 189.-Given V 4x +1+v 4x = 9; required *. 1 ✓ 3 190.-Given t? ; required x. 1-V1 - 42 1+V1 – x2 191.-A person received at one time 20 pieces of gold, 16 pieces of another value, and 12 pieces of silver for 15l. 6s. At another time 24 pieces of the first kind of gold, 30 pieces of the other, and 10 pieces of silver for 201. 195. 60.; and at another time 40 of each sort for 331. 10$. : required the value of each piece. 192.-Find three numbers on arithmetical progression so that the product of the two greatest may be 77, and the product of the two leasi 22 193.-Given x4 – 44 65 *2 - y2 = 5; required x and y. } 194.--Two workmen, A and B, were employed by the day, at different rates; A at the end of a certain number of lays had 96 shillings to receive, but B, who played 6 of those days, received only 54 shillings Now, had B worked the whole time, and A played 6 days, they would have received exactly alike. It is required to find the number of days they w re employed, and what each had per day? 195.- Find three numbers in arithmetical progression so that the sum of the two greatest may be 22 and the sum of the two least 14. 196.-A mason when he measured the length of a wall, known to be under 30 feet long, by an 8-feet rod, observed that 6 feet remained, and when he measured it by a 6-fcet rod, 4 feet remained. Required the length of the wall. 197.--To divide 100 into two such parts that their difference may be to their sum as their rectangle to the difference of their squares. 198.-Of four numbers in geometrical progression, there is given the sum of the two least = 20 = a, and the sum of the two greatest =45= b; to find the numbers. 199.- Express more simply L3a? + La4 + 5L3, where L stands for the logarithm of the expression which follows it. 200.-Given 3x3 — 6x2 – 5x – 27=0; required the value of x. LOGARITEMS. LOGARITHMs are artificial numbers, invented for the purpose of facilitating certain tedious arithmetical operations. Logaritbnis were first discovered by Jobu Napier, baron of Merchiston in Scotland, and published at Edinburgh in 1614, in his Mirifici Logarithmorum Canonis Descriptio, which contained a large canon of logarithms, with the description and uses of them; but iheir construction was reserved till the sense of the learned concerning his invention should be known. All these tables were of the kind that have since been called hyperbolical, because the numbers express the areas between the asymptote and curve of the hyperbola :-and logarithms of this kind were also soon after published by several persons; as by Ursinus in 1619, Kepler in 1624, and some others. On the first publication of Napier's logarithms, Henry Briggs, then professor of geometry in Gresham College in London, immediately applied himself to the study and improvement of them, and soon published the logarithms of the first 1000 numbers, but on a new scale, wbich he had invented; viz. in wbich the logarithm of the ratio of 10 to 1 is 1, the logarithm of the saine ratio in Napier's system being 2.30258, &c.: avd, in 1624, Briggs published his Arithmetica Logarithmica, containing the logarithms of 30,000 natural numbers, to 14 places of figures besides the index, in a form that Napier and he had agreed upon together, being the present form of logarithms. Also, in 1633, was publisher, to the same extent of figures, his Trigonometria Britannica, containing the natural and logarithmic sines, tangents, &c. In Napier's construction of logarithms, the natural numbers, and their logarithms, as he sometimes called them, or at other lines the artiticial mumbers, are supposed to arise, or to be generated, by the motions of points, describing two lines, of which the one is the natural number, and the other its logarithim, or artificial. Thus, be conceived the line or length of the radius to be described, or rua over, by a point moving along it in such a manner, that in equal portions of time it generated, or cut oft, parts in a decreasing geometrical progression, 'leaving the several remainders, or sines, in geometrical progression also; while another point described equal parts of an indefinite line, in the same equal portions of time; so that the respective sums of these, or the whole line generated, were always the arithmeticals or logarithms of the aforesaid natural sines. Briggs first adverts to the methods mentioned above, in the Appendix to Napier's Construction, wbich methods were common io both these authors, and bad, doubtless, been jointly agreed on by them. He then gives an example of computing a logarithm by the property, that the logarithm is one less than the number of places or tigures contained in that power of the given number, whose expunent is the logarithm of 10 with ciphers. Briggs next treats of the other general method of finding ihe logarithms of prime numbers, arhich he ibinks is an easier way than the former, at least when many figures are required. This method consists in taking a greater number of continued geometrical means between 1 and the given number whose logarithm is required; that is, tirst extracting the square root of the given number, then the root of the first rooi, the root of the second root, the root of the third root, and so on, till the last root shall exceed i by a very small decimal, greater or less according to the intended number of places to be in the logarithm sought; then, tinding the logarithm of this small number, by easy methods described afterwards, he doubles it as often as he made extractions of the square root; or, which is the same thing, he multiplies it by such power of 2 as is denoted by the said number of extractions, and the result is the required logarithm of the given number; as is evident from the nature of logarithms. Briggs's, or Common Logarithms, are those, therefore, that have i for the logarithm of 10, or which have 0.4342944819, &c. for the modulus; as has been explained above. Hyperbolic Logarithms are those that were computed by the inventor Napier, and called also, sometimes, natural logarithms, having 1 for their modulus, or 2.302585092994, &c. for the logarithm of 10. These have since been called hyperbolical logarithms, because they are analogous to the areas of a right-angled hyperbola between the asymptotes and the curve. Def. 1.- The base of a powe is the number from wbich the power is raised. 1 1 wef. 2.—The value of a power is the aritlimetical result, as in. dicated by that power. Cor.-Hence the value of a power does not exhibit any trace of that power. Def. 3.- The exponent of the power of a number is called the logarithm of the value of that power. Def. 4.-A system of logarithms is the numbers arising in the exponent of a power according to every value that may be given to that power, while the base is any constant number greater than unity. Def. 5. - The base of a system of logarithms is the same as the base of the power. Illustration.--Thus, let y=at, and if y be made successively equal to 1, 10, 100, 1000, 10000, &c. And the base a equal to 10, we shall bave y = ar successively, 1 = 10°, 10 = 104, 100 = 102, 1000 = 103, 10000 = 104, and so on. Then 0, 1, 2, 3, 4, are loga of 1, 10, 100, 1000, 10000, and bc. lony to a system of logarithms whose base is 10; or, if y be made suc 1 cessively 10% 100% 1000' 10000' and so on, the base being still equal to 10, 1 theo willy, which is a*, become successively = 10-1, = 10—2, 10 100 1 1 1 1 1 = 10–3, = 10-4, &c. that is 1000 10000 101'100 10000 1 &c. so that the logarithms of numbers less than unity 103' 10000 are negative. Def. 6.- When a system of logarithms has the number 2.71828, &c. for its base, that system is called the Naperian System,* and every logarithm taken from the Naperian system is called the Naperiav logarithm of its corresponding number Def. 7.—When a system of logarithms has the number 10 for its base, that system is called the Briggian System, t and every logarithm belonging to the Briggian system is called the Briggian logarithm of its corresponding number. Notation. The number 2.71828, &c. is represented boy e. rithm of any number indicated by an italic letter is represented by the corresponding capital letter. Also, the logarithm of any number, x or y, is denoted by log. x or log. y. Theorem 1.-The sum of the logarithms of any two numbers, x and 2, is equal to the logarithm of their product. For, since in the equations x = a*, and 2 = a, X is the logarithm of x, and Z is the logarithm of z, by Definition 1: multiply the corresponding sides of these two equations together, and we shall have rz = a*a? = ax+2 ; but, by Definition 3, the exponent 1 1 10 102" 1 1 104' The loga * From Napier, the inventor of logarithms. + From Henry Briggs, who was the first that changed Napier's system, by introducing 10 for the base instead of 2.71828, &c. X + Z is the logarithm of xz. Therefore, the sum of the logarithms of any two numbers is equal to the logarithms of their product. Q.E.D. Cor.-Hence it is evident that the sum of the logarithms of any number of numbers is the logarithm of the product of these numbers. Theorem 2.- The difference of the logarithms of any two numbers, x and z, is equal to the logarithm of their quotient. 3 ax For, since x = a* and 2 = 22; therefore az =a8-2; but by Definition 3, the exponent X - Z of the power of which a is the base, is the logarithm of: whence the difference of the logarithms of any two numbers is equal to the logarithm of their quotient. Q.E.D. Theorem 3.-The logarithm of the nth power of any number 1 is equal to n times the logarithm of that number. Because x = ax, take the nth power of both sides of the equa. tion, and we shall have x" = anx; but nX is the logarithm of r*; therefore the logarithm of the nth power of any number is equal to n times the logarithm of that number. Q.E.D. Theorem 4.-The logarithm of the nth root of any number x is equal to the nth part of the logarithm of that number. Because x = ax, take the nth root of both sides of this equa X tion, and we shall have x*--* : but is the logarithm of ati therefore the logarithm of the nth root of any number is equal to the nth part of the logarithm of that number. Q.E.D. Theorem 5.-The logarithm of any number, 1 + r, will be equal cer? x3 205 to the sum of the series M (x + + &c.) 3 5 For the expansion of ♡:(1 + x) or of q:(1 + 3) so as to ad 1 X n 2 4 mit of the property 9:(1+x) +9 :(1+3) = : {(1+x) * (1+z)} + ) = log. { 1 + x) * (I +)}; and, as this is included in is the series announced by this theorem; but (by Theorem 1, present article of Logarithm,) log. (1 + x) + log. (1 the functional equation, we shall then have the log. (1 + x) = M x? xt xos (or + + &c.) Q.E.D. 2 3 4 5 Cor.--Hence, if x is equal to zero, the series expressing the value of the logarithm of (1 + 2) will vanish, and the logarithm of 1 will remain : hence the logarithm of 1 is zero, or nought. |