VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 2Qg 



was above my hope, and if the screws keep an exact equality of motion forward 

 in each revolve, it is a most admirable invention ; and with the other accom- 

 modations, 1 had almost said without compare. But that the divisions of a 

 circle should be measured to seconds, without the limb of an instrument, or 

 that distances, altitudes, inclinations, and azimuths should be taken all at one 

 moment, without the limb of an instrument likewise, and each to any required 

 number of parts; or that the diameter of Jupiter should be projected in such 

 prodigious measures as you speak of, &c. were enough to amuse and amaze all 

 the mathematicians in Europe, and may indeed be rather a subject of admira-' 

 lion than belief, to any that has not known your former inventions to exceed 

 vulgar, I had almost said human, abilities. And for my part, I must confess 

 modesty so checks my ambitious desires, that I dare scarcely hope such miracles 

 should ever be produced in real practice to such exactness." Then follows an 

 account of the agreement of Mr. Horrox's Theory of the Moon with Mr. 

 Gascoigne's Observations; and also very curious ratiocinations, and a disqui- 

 sition about finding the parallax of the sun and moon, and their distance from 

 the earth. In which he censures Morinus's brags, &c. and then says, that 

 " no man that has written of the diagram (of Hipparchus) understood it fully 

 or described it rightly, but only Kepler and our Horrox; for whose immature 

 death [which was suddenly, and about the age of 25] there is yet scarcely a 

 day which I pass without some pang of sorrow." 



Thus, among many, I have related some of the passages of Mr. Gascoigne's 

 and Mr. Crabtrie's letters relating to telescopic sights. From whence it is 

 very manifest, that long before the French gentleman's claims, our countryman 

 Mr. Gascoigne had made use of those sights. in his astronomical instruments; 

 particularly in two or more sorts of micrometers, as I plainly find, and in his 

 quadrant and sextant. And had it pleased God to have given him a longer 

 life, we might have expected greater things from his pregnant and sagacious 

 wit. For he was scarcely 20 years of age when he held these correspondences 

 with Mr. Crabtrie. And at the age of 23 he was killed at Marston-Moor- 

 Battle, on July 2, l644, fighting for King Charles I. His father was Henry 

 Gascoigne, Esq. of Middleton, between Leeds and Wakefield. 



An Attempt towards the Improvement of the Method of approximating, in the 

 Extraction of the Roots of Equations in Numbers. By Brook Taylor, S.R.S. 

 N° 352, p. 6lO. 



In the Philos. Trans. N° 210, Dr. Halley has published a very compen- 

 dious and useful method of extracting the roots of affected equations of the 

 common form, in numbers. This method proceeds by assuming the root 



Q Q 2 



