What number, by taking 5 of them at a time? 8X7X6X5X4 6720 56. Ans. What number, by taking 3 of them at a time? EXAMPLE.What number of combinations can be made from 5 different things, by taking three of them at a time? What number, by taking 2 of them at a time? EXAMPLE. - Form 5 letters, a, b, c, d, e, into 10 combinations of 2 letters each; that is, into 10 unlike groups of two letters each. ab, ac, ad, ae, bc, bd, be, cd, ce, de. Ans. Form them into the greatest number of combinations possible, in collections of three each. abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde. Ans. PROBLEMS. PROB. I. The sum and difference of two numbers given, to find the PROB. II. The sum of two numbers and their product given, to find the numbers. √ (a+b)2 — (a × 6 × 4) = as b, and :b, and b+ab=a. PROB. III. The difference of two numbers and their product given, to find the numbers. N (ab)2 + (aXbX 4) = a + b, and a+b―ab PROB. IV. b, and b+ab=a. The sum of two numbers and the sum of their squares given, to find the numbers. √(a2+b2)×2— (a+b)2=ab, thence, by PROB. I. PROB. V.-The difference of two numbers and the sum of their squares given, to find the numbers. PROB. VI. √(a+b2)x2-(ab)2=a+b, thence, by PROB. I. The sum of two numbers and the difference of their squares given, to find the numbers. a2 b2 a+b =ab; 1+b=anb=b; b+anb=a; or (a+b)2 - a2 72 2 PROB. VII. — The difference of two numbers and the difference of their squares given, to find the numbers. a2 b2 a+b—ab =a+b; 2 ・=b; b+anb=a. PROB. VIII. The product of two numbers and the sum of their squares given, to find the numbers. √ (a2+b2-ax b × 2) = a b, and √ (a2+b2+axb× 2) = a+b, and PROB. IX. b, and b+ab=a. The sum of two numbers, and the product of those numbers plus the square of one of the numbers, in another sum given, to find the numbers. PROB. X.. axb+b2 a+b =b; a+b—b=a. The product of two numbers and the relation of those numbers to each other given, to find the numbers. N Xr=a, or Xr=b; r being the term in relation representing the greater number, and being the term in relation representing the less. PROB. XI. The sum of the squares of two numbers, and the relation of those numbers to each other given, to find the numbers. PROB. XII. The sum of three numbers which are in arithmetical progression, and the sum of their squares given, to find the numbers. EXAMPLE. The sum of three numbers which are in arithmetical progression is 18, and the sum of their squares is 140; what are the numbers? == 18÷3=6=b, and 18. -612 sum of a and c. (104 × 2-122) = 8 =a-c. 128: = 422c, and 2+8=10= a; the numbers, therefore, are 2, 6, and 10. Ans. NOTE. Half the sum of the first and third of three numbers forming an arithmetical progression is equal to the second number. PROB. XIII.—The sum of three numbers which are in arithmetical progression added to the sum of the greatest and least, and the sum of the squares of the numbers given, to find the numbers. 2 =a+c; thence, by PROB. XII. PROB. XIV. The sum of three numbers which are in arithmetical progression added to the sum of twice the greatest and twice the least, and the sum of the squares of the numbers given, to find the numbers. To find the altitude of an equilateral four-sided pyramid, the slant height and side of the base being known. √ (S2 — § Ã2) = h; S being the slant height, A a side of the base, and h the altitude. √ (F2 — († A)2 × 2)=h; F being the linear edge. √ (F2 — ƒ3) = h; f being half the diagonal. ✔(F2 — A × .7071′)* = h ; To find the altitude of the frustum of an equilateral rectangular 2 —a S2. pyramid. =h; S being the slant height, A a side of the greater base, and a a side of the less. F2_ ured along an angle. 2 X2=h; F being the slant height meas * Diameter X .7071 side of inscribed square. See CENTRES OF SURFACES. SECTION IV. GEOMETRY, PRACTICAL AND ILLUSTRATIVE. GEOMETRY is the science that treats of the properties of figured space. It is the science of magnitude in general, and comprehends the mensuration of solids, surfaces, lines, and their various relations. DEFINITIONS. A Point has position, but not magnitude. A Line is length without breadth, and is either Right, Curved, or Mixed. When no particular line is specified, a right line is meant. A Right Line is a straight line, or the shortest distance between two points. A Mixed Line is a right line and curved line united. Lines are parallel, oblique, perpendicular, or tangentical, one to another. An Area, surface, superfices, is the space contained within the outline or perimeter of a figure; it has no thickness, and is estimated in the square of some unit of measure, as square inch, square yard, &c. A Solid has length, breadth, and thickness, and its contents are estimated in the cube of some unit of measure. An Angle is the diverging of two lines from each other, and is right, acute, or obtuse. A Right Angle has one line perpendicular to another and resting upon it. A Triangle, or trigon, is a figure having three sides. |