11. Qu. (66) By the same. Let tangents be drawn to two circles given in magnitude and position, what is the locus of their intersection when the sum of their squares is constant? 12. Qu. (67) By Hurlothrubo Minor, of the West, In the side A. B, of a given right-angled plane triangle B A C, (A the right 4) there is a given point P; it is required to determine the point D geometrically, such, that D E being drawn perpendicular to PD, the triangle EDC may be the greatest possible. Note. This question has been proposed before, but it was only answered algebraically. 13. Qu. (68) By Mr. G. Harvey, jun. Plymouth. Two prismatic beams of equal length, whose sections are respectively a triangle and a trapezoid, the latter being cut from a triangular beam similar to the former, have their ends fixed in an upright wall, with their bases downwards; putting S, s for the strengths of the beams G, g, for the dist. of their centres of gravity from their bases, we shall have S;s::G:g; required the investigation. 14. Qu. (69) By the same. Given the difference of the segments of the base, the difference of the angles at the base, and the distance on the base between the perpendicular from the vert, angle, and point of contact of the inscribed circle: to construct the plane triangle geometrically. 15. Qu. (70) By Mr. A. Nesbit, Teacher of Mathematics, Farnley. Admit that a person be fixed vertically over FarnLEY, in lat. 53o, 47' N. at such a height as to be just able to see the sun's upper limb, at midnight on the 21st of June, should let fall a heavy body; it is required to ascertain with what velocity it would strike the earth, aud also at what distance from Farnley; the force of gravity above the earth's surface, being inversely as the square of the distance from its centre, and the earth's radius 3985 miles. 16. Qu. (71) By Mr. Jesse Winward. Given the difference of the segments of the base, made by the perp. from the vert. angle, and the difference of the angles at the base, to construct the plane triangle when the rectangle under the perpendicular, and difference of the sides, is the greatest possible. 17. Qu. (71) By Mr. A. Hirst, Marsden. There are two equal cones, the axes of each being 12 feet long, so posited that one of them standing on its base, on a horizontal plane, is just able to sustain the other, when the extremity of the base of the latter coincides with the vertex, and rests with its base on the slant side of the former; to determine the base of each cone, and also to determine, how much must be cut off from the vertex of the hanging cone, by a section perp: to its axis, so that a globe of the same matter, and whose diameter is the same as the diam. of the cone's base, may just keep the whole in equilibrio, when it is suspended at the bottom edge of the remaining frustum. 18. Qu. (72) By Mr. Vayheeg. Let tangents be drawn to two given concentric circles, to determine the locus of their intersection, when the sum of their squares is constant. 19. Qu. (73) By Mr. J. Tomlinson, Liverpool. In art. 787, Marrat's Mechanics, it is said that if the plane along which a heavy body is to be drawn be inclined to the horizon, the sine of the angle which the line of direction of the power makes with the plane, when it acts with the greatest advantage, will be ✓ (cz +9); where c is the cosine of the angle of elevation to radius unity : required the investigation. 20. Qu. (74) By Rylando. A circle and a straight line without it being given in position, it is required to determine the position of another straight line, such, that if from any point whatever in the line given by position, two tangents be drawn to the given circle, meeting the required line in с two points, the segments of the tangents intercepted between this line and the points of contact may be always equal to each other. Whoever sends the greatest number of true solutions to the preceding questions, in due time, will be entitled to a book value ten shillings and sixpence. ON THE PROPERTIES OF INFINITES AND NOTHING. the following paper, because it shows what strange things may be demonstrated, when we attempt. to proceed beyond the bounds of finite quantity. Several of your mathematical readers are well aware, that it is customary with mathematical writers to denote certain quantities by symbols, and after having come to an equation, according to the nature of the enquiry, to suppose some one of these symbols to become equal to nothing, by which means other quantities, they say, become infinitely great. These are dangerous measures, and only serve to perplex the mind with inexplicable difficulties. All quantities or magnitudes must, ultimately, be referred to numbers; for instance, if we suppose c xt=rx s, if this equation means any thing, c, t, r, and s, nust denote some numbers, let c= 2, t=0, r = 4, and s- = 3; then 2 x 6 = x 3, but if we make c or 2 = nothing, we destroy the equation, and rank nonsense is the result ; for 6 x 0, can never be equal to 4 x 3. Mr. Emerson, in his algebra, has a problem in which he proposes to explain the several properties of nothing (0) and infinity; he proceeds in the following manner. “ It is plain that nothing added to, or subtracted from any quantity, makes it neither bigger nor less. Likewise, if any quantity is multiplied by o, that is, taken no times at all, the product will be nothing." This is very plan, for o ta is = u only, and a xo is certainly equal to nothing, whatever numeral value a may be expounded by ; thus a hundred or a thousand times nothing is nothing. * Let a be 0 b 9; then if b remain the same, it is plain the less a is, the greater the quotient q will be. Let a be indefinitely small beyond all bounds, then , will be indefinitely great beyond all bounds; therefore, when a is nothing, the quotient q will be infinite. Whence also, b since = infinity, therefore, b = nothing x infinity." This is plain, the demonstration is exceedingly perspicuous, and the principles upon which it is founded are so obvious, that the conclusion must inevitably be true: what follows, it is presumed, must be equally so. Since b=0 x infinity, and since b may also be any quantity, or number, at pleasure, let b = 1; then, o x infinity =1; if b= 2, 3, 4, &c. then o x infinity = 2, 3, 4, &c. and so on for any number whatever. Therefore, it is exceedingly obvious, that o x infinity is = to any number whatever, either in the present scale of numbers, or in any other that may hereafter be invented. But further, because o x infinity = b, and b may increase beyond all bounds, in the same manner as a was supposed to decrease beyond all bounds, therefore, b may become infinitely great, that is, b may nity, and then we shall have o x infinity infinity, which is a wonderful conclusion, and proves that an infinite number of nothings is equal to infinity. Hence also, since o x infinity = infinity, we have by infinity division, 0 = or o is equal to infinity divided infinity'. by infinity ; also dividing both sides of the equation infinity equally by o, we have = infinity. Several other corollaries might be deduced. It has been proved above that o x infinity is = 1, or 2, or 3, &c. ad infinitum, we shall now demonstrate that o is, itself, = to any number whatever, that is, that 0=1,0 = 2,0 = 3, &c. Thus, the fraction , when p= 1, is evidently = 0; but by actually dividing the numerator by the be = infi. 1-P same manner denominator, it is obvious that р P- -P? =P, or when p=1, the fraction P—p = 1, therefore, 1 = 0, in the -P is, when p=l evidently equal o, р but, by division R-pi =p+p=1+1=2, when -P p=l; therefore, o= 2. In a similar way we can prove that o is 3, 4, 5, 6, 7, 8, &c. The fractions given above, by which such wonders may be performed, are called vanishing fractions; they were first brought upon the tapis a long time ago, by some French mathematicians, and produced very violent disputes among the literati of that period. Some time since, they were again brought forward by two competitors for a professorship at Cambridge, Powell and Waring : the latter, in a work published on occasion of that competition, asserted, that the fraction is = 4, when p= 1. This Powell p-PS -P = 0, and here the matter rested ; but Powell lost the professorship. It was upon the basis of vanishing fractions that Mr. Landen founded his doctrine of the “ Residual Analysis ;" hut as his doctrine had the misfortune to vanish before he died, it would not be right to conjure it again into existence. The author of a pamphlet called the Analyst, endeavoured to explain the whole mystery of vanishing fractions; he demonstrated that all such fractions are equal to nothing. “ Divide (says he) 2 - x3 by Z-X, and the quotient will be gi + 2x +xz; and supposing that z and x are equal, this same quotient will besome 3x?. But herein is a direct fallacy: for, in the first place, it is supposed that z and are unequal, without which supposition, not one step could have been made, and in the second place they are supposed denied, because, says he, when p= 1, then ? = = 2 뭐 |