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30. Given x2 + xy + y2 = 5, and x1 + x2y2 + y2 = 11, to find the values of x and y by a quadratic.
32. It is required to find by what part of the population a people must increase annually, so that they may be double at the end of every century. Ans. By 144th part nearly. 33. Required the least number of weights, and the weight of each, that will weigh any number of pounds from one pound to a hundred weight. Ans. 1, 3, 9, 27, 81. 34. It is required to find four whole numbers such, that the squares of the greatest may be equal to the sum of the squares of the other three. Ans. 3, 4, 12, and 13. 35. It is required to find the least number, which, being divided by 6, 5, 4, 3, and 2, shall leave the remainders, 5, 4, 3, 2, and 1, respectively. Ans. 59. 36. Given the cycle of the sun 18, the golden number 8, and the Roman indiction 10, to find the year. Ans. 1717. 37. Given 256x 87y= 1, to find the least possible values of x and У in whole numbers. Ans. x 52, and y = 153. 38. It is required to find two different isosceles triangles such, that their perimeters and areas shall be both expressed by the same numbers.
Ans. Sides of the one 29, 29, 40; and of the other 37, 37, 24. 39. It is required to find the sides of three right-angled triangles, in whole numbers, such, that their areas shall be all equal to each other.
Ans. 58, 40, 42;
74, 24, 70; 113, 15, 112. 40. Given x = 1.2655, to find a near approximate value
41. Given xy =
5000, and y* 3000, to find the values of Ans. x= 4.691445, and y 5.510132.
x and y.
values of x and y.
285, and y2 14, to find the Ans. x= 4.016698, and У =2.825716.
43. To find two whole numbers such, that if unity be added to each of them, and also to their halves, the sums, in both cases, shall be squares. Ans. 48 and 1680.
44. Required the two least nonquadrate numbers x and such, that x2 + y2 and x3 + y3 shall be both squares.
Ans. x 364, and y 273.
45. It is required to find two whole numbers such, that
their sum shall be a cube, and their product and quotient squares. Ans. 25 and 100, or 100 and 900, &c. 46. It is required to find three biquadratic numbers such, that their sum shall be a square. Ans. 124, 154, and 20a. 47. It is required to find three numbers in continued geometrical progression such, that their three differences shall be all squares. Ans. 567, 1008, and 1792. 48. It is required to find three whole numbers such, that the sum of difference of any two of them shall be square numbers. Ans. 434657, 420968, and 150568. 49. It is required to find two whole numbers such, that their sum shall be a square, and the sum of their squares a biquadAns. 4565486027761, and 1061652293520. 50. It is required to find four whole numbers such, that the difference of every two of them shall be a square number. Ans. 1873432, 2288168, 2399057, and 6560657.
51. It is required to find the sum of the series +
52. It is required to find the sum of the infinite series
53. Required the sum of the series 5 + 6 + 7+ 8+ 9+,
&c., continued to n terms
54. It is required to find how many figures it would take to express the 25th term of the series 21+ 2a + 2a † 2o +216 &c. Ans. 5050446 figures. 55. It is required to find the sum of 100 terms of the series (1 × 2) + (3 × 4) + (5 x 6) + (7 x 8) + (9 x 10), &c. Ans. 1343300.
56. Required the sum of 12 + 22 + 32 + 42 + 5a, &c., +50, which gives the number of shot in a square pile, the side of which is 50. Ans. 42925.
57. Required the sum of 25 terms of the series 35 + 36 × 2 + 37 × 3 + 38 × 4 + 39 × 5, &c., which gives the number of shot in a complete oblong pile, consisting of 25 tiers, the number of shot in the uppermost row being 35.
OF THE APPLICATION OF ALGEBRA TO
In the preceding part of the present performance, I have considered Algebra as an independent science, and confined myself chiefly to the treating on such of its most useful rulės and operations as could be brought within a moderate compass; but as the numerous applications, of which it is susceptible, ought not to be wholly overlooked, I shall here show, in compliance with the wishes of many respectable teachers, its use in the resolution of geometrical problems, referring the reader to my larger work on this subject for what relates more immediately to the general doctrine of curves.
For this purpose it may be observed, that when any proposition of the kind here mentioned is required to be resolved algebraically, it will be necessary, in the first place, to draw a figure that shall represent the several parts, or conditions, of the problem under consideration, and regard it as the true one.
Then, having properly considered the nature of the question, the figure so formed, must, if necessary, be still farther prepared for solution, by producing, or drawing, such lines in it as may appear, by their connexion or relations to each other, to be most conducive to the end proposed.
This being done, let the unknown line, or lines, which it is judged will be the easiest to find, together with those that are known, be denoted by the common algebraical symbols, or letters; then, by means of the proper geometrical theorems, make out as many independent equations as there are unknown
* The learner, before he can obtain a competent knowledge of the method of application abovementioned, must first make himself master of the principal propositions of Euclid, or of those contained in my Elements of Geometry; in which work he will find all the essential principles of the science comprised within a much shorter compass than in the former.
And in such cases where it may be requisite to extend this mode of application to trigonometry, mechanics, or any other branch of mathematics, a previous knowledge of the nature and principles of these subjects will be equally necessary.
APPLICATION OF ALGEBRA TO GEOMETRY.
quantities employed; and the resolution of these, in the usual manner, will give the solution of the problem.
But as no general rules can be laid down for drawing the lines here mentioned, and selecting the properest quantities to substitute for, so as to bring out the most simple conclusions, the best means of obtaining experience in these matters will be to try the solution of the same problem in different ways; and then to apply that which succeeds the best to other cases of the same kind, when they afterwards occur.
The following directions, however, which are extracted, with some alterations, from Newton's Universal Arithmetic and Simpson's Algebra and Select Exercises, will often be found of considerable use to the learner, by showing him how to proceed in many cases of this kind, where he would otherwise be left to his own judgment :
1st. In preparing the figure in the manner abovementioned, by producing or drawing certain lines, let them be either parallel or perpendicular to some other lines in it, or be so drawn as to form similar triangles; and, if an angle be given, let the perpendicular be drawn opposite to it, and so as to fall, if possible, from one end of a given line.
2d. In selecting the proper quantities to substitute for, let those be chosen, whether required or not, that are nearest to the known or given parts of the figure, and by means of which the next adjacent parts may be obtained by addition or subtraction only, without using surds.
3d. When in any problem there are two lines, or quantities, alike related to other parts of the figure, or problem, the best way is not to make use of either of them separately, but to substitute for their sum, difference, or rectangle, or the sum of their alternate quotients; or for some other line or line in the figure, to which they have both the same relation.
4th. When the area, or the perimeter, of a figure is given, or such parts of it as have only a remote relation to the parts that are to be found, it will sometimes be of use to assume another figure similar to the proposed one, that shall have one of its sides equal to unity, or to some other known quantity; as the other parts of the figure, in such cases, may then be determined by the known proportions of their like sides, or parts, and thence the resulting equation required.
These being the most general observations that have hitherto been collected upon this subject, I shall now proceed to eludicate them by proper examples; leaving such farther remarks as may arise out of the mode of proceeding here used, to be applied by the learner, as occasion requires, to the
solutions of the miscellaneous problems given at the end of the present article.
PROBLEM 1.-The base, and the sum of the hypothenuse and perpendicular of a right-angled triangle being given, it is required to determine the triangle.
Let ABC, right-angled at c, be the proposed triangle; and put BC = b, and AC = x.
Then, if the sum of AB and AC be represented by s, the hypothenuse AB will be expressed by s-x.
But, by the well-known property of right-angled triangles (Euc. 1, 47,)
AC2 + BC2 x2 + b2 = så
Whence, omitting a2, which is common to both sides of the equation, and transposing the other terms, we shall have 28x 52 b2, or
which is the value of the perpendicular Ac; where s and b may be any numbers whatever, provided s be greater than b. In like manner, if the base and the difference between the hypothenuse and perpendicular be given, we shall have,, by putting x for the perpendicular and d +x for the hypoth
Where the base (b) and the given difference (d) may be any numbers as before, provided b be greater than d.
PROBLEM 2.-The difference between the diagonal of a square and one of its sides being given, to determine the
* The edition of Euclid referred to in this and all the following problems, is that of Dr. Simson, London, 1801; which may also be used in the geometrical construction of these problems, should the student be inclined to exercise his talents upon this elegant but more difficult branch of the subject. The New York edition of Brewster's Legendre, or Playfair's Euclid, will answer the same purpose.