in all 4 ways. Just so each of the above 6 gives 4 ways, therefore in all 4 × 6=24=1×2×3× 4. By observing the several series of numbers, which we have found, we may determine the general rule. On the addition of each successive point, the number of the preceding ways has been multiplied by the number which denotes the present number of points. Therefore to find the number of different combinations which can be made with any number of points; Multiply together the natural series of numbers from one up to, and including, that number which denotes the number of the given points. 22. What is the greatest number of points at which any given number of straight lines may intersect one another? Answer. 2 straight lines can only touch each other at one point, and this point is common to the 2 lines. 3 straight lines. The 3d line may intersect each of the preceding; and thus we have 2 points more; this gives in the whole 1+2=3 points. 4 straight lines. The 4th line may intersect each of the other 3, and thus we have 3 new points; 1+2+3 6 points, (fig. 14.) = In a similar manner the 5th line may intersect each of the other 4 lines, and in general each successive line may intersect all which precede it. Thus with each successive line there will be as many new points as there were lines already. The 6th line gives 5 new points. " 6 66 7th " 100th " Rule. To find the greatest number of points of intersection, which a given number of straight lines may make: add together the natural series of numbers from 1 up to, but without including, that number which makes the number of lines. We can arrive at the same result by a shorter process. It is evident that straight lines may be drawn in such a manner, that each line shall intersect all the others. Take, for example, 20 lines; each of these may intersect the other 19. Thus there will be 19 points in each line; this would give in all the lines 19×20=380 : but each point is common to two lines; we must therefore divide this product by 2; thus for 20 lines we have 20 X 19 190. Thus we get another rule; Multiply the number of lines by the same number less 1, and divide this product by 2. 2 = 23. We will now reverse the operation. Let us suppose the greatest number of points at which a certain number of straight lines can intersect one another to be 45; what is the number of lines? Answer. We have already learnt that the number of intersection points is the half of the product of two numbers differing from each other only by 1. We must therefore multiply the given number by 2, and then seek for two numbers, differing from each other only by 1, which multiplied together will give this product. For example, 45 × 2=90=10 × 9; therefore the number of lines which will give 45 points of intersection is 10. 24. The intersecting straight lines may be divided into many sets. We will first suppose them to be divided into 2 sets, and will consider several different cases. 1. The lines of each set parallel among themselves. Example. Six lines are divided into 2 sets, one of which has 4 lines and the other 2 lines; the 4 are parallel one to the other, and the 2 are parallel to each other. What is the greatest number of intersection points? Answer. Each line of one set intersects each line of the other set, thus we have 4 × 2=8. Ten straight lines may be divided into 2 sets as follows: 9 and 1, which gives 9 x 19 points. Observe the series of numbers, 9, 16, 21, 24, 25; the differences between the successive numbers form the series 7, 5, 3, 1. 2. The lines of each set intersecting among themselves at one point, as in this figure. (Fig. 15.) Here the lines of each set give one intersection point among themselves. The lines of one set also intersect the lines of the other. Multiply the number of lines in one set by the number of lines in the other set, and add 2 to the product; we shall thus have the number of intersection points. The number 12 being divided into 10 and 2 gives (10 × 2)+2=22 points. 3. The lines of one set parallel to one another, and the lines of the other set intersecting one another at one point. In this case each line of the second set intersects each line of the first set, and the lines of the second set give one intersection point besides. To ascertain the whole number of points, we must multiply the numbers of the two sets together and add 1 to the product. Example. Divide the number 16 into 2 sets as follows, the 1st column being the number of lines in the set of parallels. 14 and 2 give (14 × 2)+1=29 The series of numbers, 29, 40, 49, 56, 61, 64, 65, gives the series of differences 11, 9, 7, 5, 3, 1. 4. The lines of one set parallel, and the lines of the other set intersecting one another at the greatest number of points. Each line of the second set intersects each line of the first set; the number of points at which the lines of the 2d set intersect one another must be calculated by the rule before given, (22.) The number 20, for example, being divided The series of numbers, 37, 54, 70, 85, 99, 112, 124, 135, 145, gives a series of differences 17, 16, 15, 14, 13, 12, 11, 10. By continuing the preceding calculations, (1, 2, 3,) a series of differences is obtained, the reverse of those already obtained. By continuing the calculation for lines divided into sets in this manner, (4, 5,) the series simply progresses. 5. The lines of one set intersecting one another at the greatest number of points, and the lines of the other set intersecting one another at one point. In this case, first calculate the number of points in the first set; then multiply the number of lines of the 2d set by the number of lines in the 1st set, because each line of the one set intersects each line of the other set, and add 1 to the product, because the lines of the 2d set also |