e, d, c being so many and so placed that the rectangles bk ci, ei, fb taken together shall differ less from fghikmb than d does: all which may be easily done by the help of the equation of the curve, and the difference between D and the figure fghikmb given. Then since di is the longest of the perpendicular ordinates that insist upon fb, the rest will gradually decrease as they are farther and farther from it on each side, as appears from the construction of the figure and consequently eb is greater than gf or any other ordinate that insists upon eJ Now if Ao were equal to Ae, then by lem. 2. the probability of the event Min a single trial would be the ratio of Ae to AB, and consequently by cor Prop. 1 the probability of it's failure would be the ratio of Be to AB. Wherefore, if r and r be the two forementioned ratios respectively, by Prop. 7. the probability of the event M happening p times and failing q in p+ g trials would be e rPr. But a nd r being respectively the ratios of Ae to AB and Be to AB, if y is the ratio of eb to AB, then, by construction of the figure aib, y= ExPrq. Wherefore, if Ao were equal to Ae the probability of the event M happening p times and failing q times in p+g trials would be y, or the ratio of eb to AB. And if Ao were equal to Af, or were any mean between Ae and Af, the last mentioned probability for the same reasons would be the ratio of fg or some other of the ordinates insisting upon ef, to AB. But eh is the greatest of all the ordinates that insist upon ef Wherefore, upon supposition the point should lie any where between f and e, the probability that the event M happens p times and fails g in p+g trials cant be greater than the ratio of eh to AB. There then being these two subsquent events. the lst that the point o will lie between e and f, the 2d that the event M will happen p times and fail g in p+g trials, and the probability of the lst (by lemma lst)is the ratio of ef to AB, and upon supposition the lst happen by what has now been proved, the probability of the 2d cannot be greater tha the ratio of eh to AB it evidently follows(from Prop. 3. that the probability both together will happen cannot be greater than the ratio compounded of ef to AB and that of eh to AB, which compound ratio is the ratio of fh to CA. Wherefore, the probability that the point o will lie between f and e, and the event M will happen p times and fqil g, is not greater than the raio of fh to CA. And in like, manner the probability the point o will lie between e and and the event M happen and fail as before, cannot be greater than the raio of ei to CA. And again, the probability the point o will lie between c and b, and the event M happen and fail as before, cannot be greater than the ratio of bk to CA. Add now all these several probabilities together, and their sum(b Prop. 1)will be the probability that the point will lie somewhere between f nd b, and the event M happen p times and fail q in p+q trials. Add likewise he correspondent ratios together, and their sum will be the ratio of the sum of the antecedents to their consequent, i. e. the ratio of fb, ei, ci, bk together CA; which ratio is less than that of d to CA, because D is greater than fh, ci,bk together. And therefore, the probability that the point o will lie between f and b, and withal that the event M will happen p times and fail g in p+q times, is less than the ratio of d to CA; but it was supposed the same which is absurd. And in like manner, by inscribing rectangles within the figure, as eg,e, d, c being so many and so placed that the rectangles bk, ci, ei, f b taken together shall differ less from fghikmb than D does; all which may be easily done by the help of the equation of the curve, and the difference between D and the figure fghikmb given. Then since di is the longest of the perpendicular ordinates that insist upon f b, the rest will gradually decrease as they are farther and farther from it on each side, as appears from the construction of the figure, and consequently eb is greater than gf or any other ordinate that insists upon ef. Now if Ao were equal to Ae, then by lem. 2. the probability of the event M in a single trial would be the ratio of Ae to AB, and consequently by cor. Prop. 1. the probability of it’s failure would be the ratio of Be to AB. Wherefore, if x and r be the two forementioned ratios respectively, by Prop. 7. the probability of the event M happening p times and failing q in p + q trials would be E xprq. But x and r being respectively the ratios of Ae to AB and Be to AB, if y is the ratio of eb to AB, then, by construction of the figure AiB, y = Exprq. Wherefore, if Ao were equal to Ae the probability of the event M happening p times and failing q times in p+q trials would be y, or the ratio of eb to AB. And if Ao were equal to Af, or were any mean between Ae and Af, the last mentioned probability for the same reasons would be the ratio of fg or some other of the ordinates insisting upon ef, to AB. But eh is the greatest of all the ordinates that insist upon ef. Wherefore, upon supposition the point should lie any where between f and e, the probability that the event M happens p times and fails q in p+q trials can’t be greater than the ratio of eh to AB. There then being these two subsquent events. the 1st that the point o will lie between e and f, the 2d that the event M will happen p times and fail q in p + q trials, and the probability of the 1st (by lemma 1st) is the ratio of ef to AB, and upon supposition the 1st happens, by what has now been proved, the probability of the 2d cannot be greater than the ratio of eh to AB it evidently follows (from Prop. 3.) that the probability both together will happen cannot be greater than the ratio compounded of that of ef to AB and that of eh to AB, which compound ratio is the ratio of fh to CA. Wherefore, the probability that the point o will lie between f and e, and the event M will happen p times and fqil q, is not greater than the raio of fh to CA. And in like, manner the probability the point o will lie between e and d, and the event M happen and fail as before, cannot be greater than the raio of ei to CA. And again, the probability the point o will lie between c and b, and the event M happen and fail as before, cannot be greater than the ratio of bk to CA. Add now all these several probabilities together, and their sum (by Prop. 1.) will be the probability that the point will lie somewhere between f and b, and the event M happen p times and fail q in p + q trials. Add likewise the correspondent ratios together, and their sum will be the ratio of the sum of the antecedents to their consequent, i. e. the ratio of f b, ei, ci, bk together to CA; which ratio is less than that of D to CA, because D is greater than fh, ei, ci, bk together. And therefore, the probability that the point o will lie between f and b, and withal that the event M will happen p times and fail q in p + q times, is less than the ratio of D to CA; but it was supposed the same which is absurd. And in like manner, by inscribing rectangles within the figure, as eg, 10