68 PHILOSOPHICAL TRANSACTIONS. [aNNO 1703. 



prehend all the cases of the problem, where 3 points are given in any line, and 

 a 4th is required such, that the rectangle under the segments of the proposed 

 line intercepted between the point sought, and two of the given points shall 

 bear a given ratio to the square of the segment terminated by the third point. 

 The cases indeed of the problem, from the diversity of situation in the points 

 given to the points sought, and to one another, are in number 6. The given 

 extreme of the segment to constitute the square, may either be without the other 

 2 given points, or between them. And when it is without, the point sought 

 may be required to be taken without them all, either on the side opposite to the 

 given extreme of the segment to constitute the square, which will be one case, 

 or it may be required to fall on the same side, which will be a second case. If 

 it be required to fall between this point and the other two, this will be a 3d case. 

 A 4th case will be, when the point sought shall be required to fall between the 

 other two points. Also when the given extreme of the segment to constitute 

 the square lies between the other two given points, the point sought may be re- 

 quired to fall, either there also, or without, composing the 5th and 6th cases. 



The propositions in Pappus referring to these cases, though but 4 in number, 

 suffice for them all, each proposition being applicable to the problem 2 ways. 

 For instance, the 35th prop, as expressed by Pappus, is this, being the first 

 above cited. Three points c, d, e being taken in the line ab, so that the 



rectangle under abe be equal to that under cbd, ab is to be as ~'~ '~^ J^ 

 the rectangle under dac to that under ced. Now ab is to be, both as the 

 square of ab to the rectangle under abe, and as the rectangle under abe to the 

 square of be. Therefore, if the ratio of ab to be be given, the ratio of the 

 square of ab to the rectangle under cbd will be given, which is the first of the 

 cases above described, and also the ratio of the rectangle under cbd to the square 

 of BE given, which is the 2d case. In both cases, the rectangle under dac will 

 be to that under ced, in the given ratio of ab to be. But in the first the rect- 

 angle under dac will be given, and the point e in the rectangle under ced to be 

 found by applying a rectangle, which shall bear a given ratio to the given rect- 

 angle under dac, to the given line cd exceeding by a square ; and in the 2d case 

 the rectangle under ced is given, and a in the rectangle under dac to be found, 

 by applying to the given line cd a rectangle exceeding by a square, which shall 

 bear a given ratio to the rectangle under ced now given ; whence, by the ratio 

 of AB to BE given, the point b will be found in both cases. The 22d prop, 

 either way applied refers to the 3d case only ; the 30th relates both to the 4 th 

 and 5th ; and the 36th prop, to the remaining 6th. 



The 45th, and other following propositions, are accommodated' to the solu- 

 tion of Apollonius's problem, wiien 4 points are given, and a fifth required. 



