108 



P HI S M S. 



Purisms, is unquestionably rather obscure; and without an ex- 

 S -*""V" < *' > ' ample of one of these propositions, it is by no means 

 easy to comprehend its meaning: it appears there- 

 fore preferable to postpone the explanation of the 

 term until the reader is made acquainted with the 

 thing. The ancient geometers examined every prob- 

 lem on which they bestowed their attention with the 

 most minute scrutiny: unacquainted with the com- 

 prehensive generalization which is introduced into 

 every geometrical problem by the application of alfre- 

 bra, they carefully inquired into every separate case 

 that could cause any change in the magnitude or rela- 

 tive position of the data, fearful lest that mode of so- 

 lution they had contrived for it in one case, might not 

 equally apply to others. Such a laborious course of 

 inquiry, iilthough adverse to rapid advancement, was 

 well calculated to make them perfectly acquainted 

 with every thing remarkable which the solution of the 

 problem could present ; and it must soon have occur- 

 red to them, that in many cases the general construc- 

 tion would fail, and no solution be obtained, in conse- 

 quence of some peculiar relation between the data. 

 Such is the case if we attempt to divide a given line 

 into two parts, whose rectangle is equal to a given 

 square. When the given square is greater than that 

 described on half the given line, no solution can be ob- 

 tained. In such cases, the problem became impossible, 

 and it was always found that some two at least of the 

 data were contradictory to each other. In the illus- 

 tration, we have chosen the two conditions, defining 

 the magnitude of the line and that of the rectangle of 

 its segments are incompatible. 



When a problem contained an impossible case, ano- 

 ther question presented itself; to determine the limits 

 amongst the relations of the data, so that it shall just 

 remain possible ; and with respect to the problem it- 

 self, to construct it so that a certain quantity, instead 

 of being given, shall be the greatest or least possible. 

 The elegant constructions to which this gave rise, un- 

 der the name of maxima and minima, are well known 

 to geometers. 



These circumstances would occur when the data 

 were but few, and the problem simple ; but in the 

 consideration of questions a little less elementary, it 

 must have been observed, that besides this method, by 

 which the construction became useless, another of quite 

 an opposite nature was sometimes introduced. It might 

 happen that two lines or two circles by whose inter- 

 section the point ought to be determined, instead of 

 cutting each other as in the general case, or not inter- 

 secting each other at all, as in the impossible one, 

 should wholly coincide. The true interpretation of 

 this circumstance could not long remain unnoticed. 

 Since that point, which was common to the two lines, 

 determined the point to be found, and in the case of 

 two circles intersecting, there were two points in com- 

 mon, and therefore equally fulfilling the condition, it 

 was natural to conclude, that when the lines or circles 

 coincided, all points being in common, all would equal- 

 ly satisfy the problem. Here, then, an infinity of so- 

 lutions appeared, yet they were all connected by a cer- 

 tain law. The reason of such a singular result, must 

 soon have been found in the coincidence between two 

 of the data, and thus a less number of data being given 

 than were sufficient, the problem became indetermi- 

 nate. 



These curious cases would, of course, become ob- 

 jects of research, from the great facilities they afforded 

 for the solutions of the problems to which they belong- 



ed, and the elegance which they introduced into them ; 

 and partaking in some measure of the nature of prob- 

 lems, as well as of theorems, the\yformed an interme- 

 diate class of propositions of great importance, to which, 

 when enunciated in a peculiar manner, the name of 

 porisms was attached. 



As an example of the manner in which a porism 

 might be discovered, we shall consider the following 

 problem. 



A circle ABC, Plate CCCCLVII. (Fig. l.)a straight 

 line DE, and a point F, being given in position, to find a CCCCLXVII. 

 pomtGin the straight line DE, such that GF, the line ^'g- 1- 

 drawn from it to the given point, shall be equal to GB, 

 the line drawn from it touching thf given circle. 



Suppose the point G to be found, and GB to be 

 drawn touching the circle ABC in B ; let H be the 

 centre of the circle ABC ; join HB, and let HD be 

 perpendicular to DE ; and from D draw DL, touch- 

 ing the circle ABC in L, and join HL. Also, from the 

 centre G, with the distance GB or GF, describe the 

 circle BKF, meeting HD in the points K and K'. 



It is plain that the lines HD and DL are given in 

 position and in magnitude. Also, because GB touches' 

 the circle ABC, HBG is a .right angle, and G is the 

 centre of the circle BKF; therefore HB touches the 

 circle BKF, and consequently the square of HB or of 

 HL is equal to the rectangle K'HK. But the rectangle 

 K'HK, together with the square of DK, is equal to 

 the square of DH, because KK' is bisected"in D ; there- 

 fore the squares of HL and DK are also equal to the 

 square of DH. But the squares of HL and LD are 

 equal to the square of DH ; wherefore, the square of + 



DK is equal to the square of DL, and the line DK to 

 the line DL. But DL is given in magnitude, there- 

 fore DK is given in magnitude, and K is therefore a 

 given point. For the same reason, K' is a given point, 

 and the point F being also given by hypothesis, the 

 circle BKF is given in position. The point G, therefore, 

 the centre of the circle BKF, is given, which was to be 

 found. 



Hence this construction : Having drawn HD per- 

 pendicular to DE, and DL touching the circle ABC, 

 make DK and DK' each equal to DL, and find by the 

 centre of a circle described through the points K, F and 

 K', that is, let FK' be joined, and bisected at right 

 angles by the line MN, which meets DE in G ; G will 

 be the point required, or it will be such a point, that 

 if GB be drawn from it, touching the circle ABC, and 

 GF to the given point, GB and BF will be equal to one 

 another. 



In this instance, we have a problem which admits in 

 general but of one solution, since only one circle can 

 pass through three given points; yet if the point F 

 which is given should coincide with either of the two 

 points K or K' which are found, it is evident that an 

 infinite number of circles can pass through two given 

 points, and their centres will be situated on a right line 

 perpendicular to the middle point of the line which 

 joins them : in this case, then, the problem becomes 

 indeterminate, since any point in that line will satisfy 

 the conditions. 



The indeterminate case is thus enunciated as a po- 



rism. 



A circle ABC being given by position, and also a 

 straight line DE, which does not cut the circle, a point 

 K may be found such, that if G be any point whatever 

 in the line given, the straight line drawn from G to 

 the point K shall be equal to the straight line drawn 

 from G touching the circle ABC. This is in fact the 



