60 PHILOSOPHICAL TRANSACTIONS. [aNNO 1763. 



correction,) and that the time of the internal contact should have been observed 

 at 21'' 40"' 52 ; the parallax by taking a mean will then be Q".732, exactly 

 agreeing with a mean of all the other determinations. And in this quantity of 

 the sun's parallax we must either acquiesce, or remain as ignorant of the true 

 quantity of it as we were before, till we can have recourse to the next transit on 

 June 3d, 1769, when the planet Venus will again pass over the sun's disk, having 

 something more than 10 minutes of north latitude; and will be so favourably cir- 

 cumstanced, that if the errors in observing each contact do not exceed 4" or 3% the 

 quantity of the sun's parallax may be determined within less than -^-^ part of the 

 whole ; as the total duration, or the interval between the two internal contacts, 

 will be found to be about 1 8 minutes longer at Tornea° than at Mexico. But 

 the several circumstances of that transit must be the subject of a future paper. 

 Let it suffice at present to observe, that it will in part be visible to the inhabitants 

 of this island, as Venus will be seen wholly entered on the sun's disk more than 

 half an hour before the time of sun-set at Greenwich. 



LFI. On the Locus for Three and Four Lines, celebrated among the Ancient 

 Geometers. By H. Pemberton, M. D., R. S. Land, et R. A. Berol. S. p. 496. 



My worthy friend, and associate in my early studies, the collector* of the late 

 Mr. Robins's mathematical tracts, thought it conducive to a more complete vin- 

 dication of the memory of his friend against an insinuation prejudicial to his can- 

 dour, to make some mention of the course I took in my early mathematical 

 pursuits, and how soon I became attached to the ancient manner of treating geo- 

 metrical subjects. This gave occasion to my looking into some of my old papers, 

 among which I found a discussion of the problem relating to the locus ad tres et 

 quatuor lineas, celebrated among the ancients, which I then communicated to a 

 friend or two, whose sentiments of those ancient sages were the same with mine. 

 What I had drawn up on this subject is contained in the papers following. 



The describing a conic section through the angles of a quadrilateral with two 

 parallel sides, is so ready a means of assigning loci for the solution of solid pro- 

 blems, that it cannot be doubted but this gave rise to the general problem con- 

 cerning 3 and 4 lines mentioned by Apollonius, and described by Pappus; and it 

 may be learned from Sir Isaac Newton, who has considered the problem, how 

 easily the most extensive form of it is reducible to the case which probably gave 

 rise to it. 



Sir Isaac Newton refers the general problem to this: any quadrilateral abcd, 

 (fig. 1, pi. 2) being proposed, to find the locus of the point p, whereby peu 

 being drawn parallel to AC, and spt parallel to ab, the ratio of the rectangle 



* * IJr. James Wilson. 



