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150.- In a right angled triangle, there is given the perpendicular = 30. It is required to find the base and area, when the reciangle under the difference of the sun of the legs and hypothenuse, and the difference of the legs is a minimum.
151.-With any number a of cards, suppose that I form any number b of beaps, each containing any equal number c of points. These points being reckoned on the supposition that, if the first card of any heap be an ace, it shall tell for 11 points; if a figured card, as a king, queen, or knave, it tells for ten points; and if any other card, fur the number upon it, as 9 for a 9 of any sort, &c. the remaining cards of the same leap tell for only one point each. After all the heaps are formed, I place in your hand the number d of cards which remain, and require you to divine the total number of points formed by the first cards only belonging to each licap. Iu what manner will you ascertain that number?
152.-Required a number surb that, if it be divided into m equal parts, the continual product of all these parts shall be equal to the continual product of m + 1 equal parts of the same number.
153.-Suppose on AB, the horizontal line, a cannon was discharged at A; it is required to find how high in the air BC must be raised, that Jetting fall a ball, it shall arrive at the ground (B) the same instant (of time) the sound arrives at c, but not till of a second after the sound arrives
& at B.
154.-A gentleman bas a parabolic garden, whose area = 16 acres, also within the said garden is a right-angled triangle fish-pond, formed by the semi-ordinate part of the axis, and a right line drawn from the centre of gravity terminating in the extremity of the semi-ordinate. Now, the area of this right-angled triangle is equal to its perimeter. Query, the axis and ordinate of this parabolic garden, as also the area and sides of the triangular pond'
155.- What difference is there between a floor 48 feet long, and 30 feet broad, and two others each of half the dimensions ?
156.--From a mahogany plank 26 inches broad, a yard and a ball is to bc sawed off'; at what distance from the end must the line be struck ?
157.-The sides of 3 squares being 4, 5, and 6 feet, respectively, it is required to find the side of a square that shall be equal in area to all the three.
158.-The captain of a privateer descrying a trading vessel 7 miles a-head, sailed 20 miles in direct pursuit of bir, and, then observing the trader steering in a direction perpendicular to her former course, changed his own course so as to overiake her without making another tack. On comparing their reckonings it was found, tbat the privateer had run at the rate of ten knois in an hour, and the trading vessel at the rate of eight knots in the same time. Required the distance sailed by the privateer. 159.--Given a +* + V 2ax + xx
b., required x. at*-V 2ax + xx 160.--To find that number, which being divided into either 3 or 4 equal parts, the continaal product of all the parts shall in both cases be exactly the same.
161.-A gentleman has a garden in ihe form of a quadrant, whose radius is 40 poles; in which he bas ordered his gardiner to make a canal at olle of its vertices, in such sort that the rectangle of the secant and co sine sball exceed the product of the cotangent and versed sine by a maximum ; required the garden's area.
; 162.-In what north Jatitude is the shortest day equal şof of the longest at London?
163.-To find two numbers in the proportion of 9 to 7, such that the square of their sum shall be equal to the cube of their difference.
164.—To divide 36 into 3 such parts, that I of the first, $ of the seconil, and 4 of the third, may be equal to each.
165.—. To divide the number 90 into 4 such parts that if the first be increased by 5, the second diminished by 4, the third multiplied by 3, and the fourth divided by 2, the result in each case shall be exactly the same.
166.—Three persons having bought a sagar-ral, want to divide it equally among them, by sections parallel to the base; it is required to fiod the altitude of each person's share, supposing the loaf to be a cone, whose beight is 20 inches.
167.-How bigh above the surface of the earth must a person be raised to see a third part of its surface?
168.—A cubical foot of brass is to be drawn into a wire of 2 of an inch in diameter; what will be the length of the wire, allowing no loss in the metal.
169.-One day, in my rounds, I took the following dimensions of a conical vesscl on purpose to puzzle a fellow officer, viz. the top diameter 100 inches, then down the slant side 30 inches, from which point I took the diagonal to the bottom 120 inches, and then the bottom diameter 80 inches. Required the content of the vessel.
170.-A gentleman bonght the right of enclosing two felds, the one is to be in the form of a circle, and the other in that of an ellipse, whose two axes are as 3 to 2, the sum of the greater arc, and the diameter of the circle is to be 120 poles, also for every square pole, enclosed in the circular field, he is to pay 6 shillings, and for the same quantity in the other, 4 shillings, I desire to know the content of each field, supposing he hath the least ground possible for his money.
171.–To find that number whose square root is to its cube root in the proportion of 5 to 2.
172.-A grocer with 56lbs. of fine tea, at 20 shillings per pound, would mix a coarser sort at 14 shillings, so as to afford the whole together at 18 shillings per pound. What quantity of the latter sort must he take?
173.- Find a simple logarithmic expression for 23 V 23.
174.–To determine tho latitude, N., day of the montlı, and hour of the day, when the sun is due east, and the rectangle under the cube of the sino and sqnare of the cosine of the sun's altitude, a maximum; the sun's declination being eqnal to one-third of his altitude at that time.
175.-To determine the dimensions and area of a plane triangle, when the sum of the squares of the greater segment of the base and perpendicular is = 100 chains, the ratio of the segments of the base, as 3 to 2, and the area a maximum.
IV 5+ y = 3v 176.-Giren,
; fo find the values of v and X.; !+ V x 177.-ljiven the perimeter of an ellipso = 100 poles, and the rectangle
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 as + ms divided by a + m. 181.- What is the quotient of ab - mo divided by a t m? 182.–Required the quotient of al — ms divided by a - 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 + y3 : *3 -- y3 :: 234 : 109
}; required x and y.
✓ (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 remaining solid. 189.-Given V 4x+1+4x
= 9; required *.
; required x. 1-V1 *2 1+v 191.- A person received at one time 20 pieces of gold, 16 pieces of another value, and 12 pieces of silver for 151. 6s. At another time 24 pieces of the first kind of gold, 30 pieces of the other, and 10 pieces of silver for 201. 19s. 6d.; and at another time 40 of each sort for 331. 10s. : required the value of each piece.
i92.- Find three numbers on arithmetical progression so that the product of the two greatest may be 77, and the product of the two least 22. 193.-Given x4 - 44 = 65 ) *2 - y2 :
5}; required x and yo
194.---Two workmen, A and B, were employed by the day, at different rates; A at the end of a certain number of days bad 96 shillings to receive, but B, who played 6 of those days, received only 54 shillings, Now, bad 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 wre 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 L3a2 + La+ + 513, where L stands for the logarithm of the expression which follows it.
200.-Given 3x3 — 6x2 – 5x – 27=0; required the value of x.
LOGARITHMS. LOGARITH Ms 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 bis invention should be known).
All these tables were of the kind that have since been called hyperbolical, because the numbers espress 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, which he had invented; viz, in which the logarithm of the ratio of 10 to 1 is 1, the logarithm of the same ratio in Napier's system being 2.30258, &c.: and, 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 1033, was publisherl, 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 thines the artificial numbers, 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 logarithm, or artificial. Thus, be conceived the line or length of the radius to be described, or rud 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 geoimetrical 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, which methods were common to both these authors, and had, doubtless, been jointly agreed on by them. lle then gives an example of computing a logarithm by the property, that the logarithm is one less than the number of places or figures contained in that power of the given number, whose expopent is the logarithm of 10 with ciphers. Briggs next treats of the other general method of finding the logarithms of prime numbers, which he thinks 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 nuinber whose logarithm is required; that is, first extracting the square root of the given number, then the root of the first rool, the rooi of the second root, the root of the tbird root, and so on, till the last root shall exceed 1 by a very small decimal, greater or less according to the intended nuinber 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, be 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 beer 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.