vol,. XLVI.] PHILOSOPHICAL TRANSACTIONS. 6Q 



On the late Dr. Hnlley's Demonstration of the Analogy of the Logarithmic 

 Tangents to the Meridian Line, or Sum of the Secants. By Mr. John Robert- 

 son* F.R.S. N"496, p. 559. 



Dr. Halley, in this tract, N" 219, Philos. Trans, seems to have had 2 points 

 chiefly in view, first, to prove, that the divisions of the meridian line in a Mer- 

 cator's chart, were analogous to the logarithmic tangents of the half-complements 

 of the latitudes. 2dly. To find a rule by which the tables of meridional parts 

 might be computed from Brigg's, or the common logarithmic tangents. The 

 former of these the Doctor has clearly and elegantly proved, but lie has given 

 rather too few steps to show as clearly the investigation of the latter. 



Article 1 . If the circumference of a circle be divided into any number of equal 

 parts, by as many radii; and a line be drawn from the circumference cutting those 

 radii, so that their parts intercepted between this line and the centre be in a 

 continued decreasing geometric progression ; then will that intersecting line be a 

 curve, called the proportional spiral, and will intersect those radii at equal angles. 

 This will be evident, by supposing the radii so near to each other, that the inter- 

 cepted parts of the spiral may be taken as right lines; for then there will be a 

 series of similar triangles, each having an equal angle at the centre, and the sides 

 about those angles proportional. 



2. The same things still supposed: the parts of the circumference of the circle 

 reckoned from any one point, may be taken as the logarithms of the ratios be- 



* John Robertson, f.r.s. the author of this paper, was born in the year 1712; and though he 

 was at first placed out in a trade, yet he must soon have quitted it, as in the title of his first book, a 

 Complete Treatise on Mensuration, in 1739, he is stiled teacher of the mathematics. In this line, 

 as a private teacher, he continued several years, till in 1754 he was appointed master of the Royal 

 Mathematical School in Christ's Hospital ; in which year also he published the first edition of his 

 Elements of Navigation. The year following however he left Christ's Hospital, in consequence of 

 an Admiralty appointment to be first master of the Royal Naval Academy at Portsmouth; soon after 

 which, he published his treatise on mathematical instruments. In 1766, through the petty cabals of 

 the second master, he was dismissed from his situation by the first lord of the Admiralty; on which 

 he returned to London, where, in the latter part of that year, or early in the next, l)e was appointed 

 clerk and librarian to the Royal Society; an employment which he respectably held to the time of 

 his death, in Dec. 1776, at 64 years of age. 



Besides the three publications above-mentioned, which were all excellent of their kind, particu- 

 larly the navigation, and have gone through numerous editions ; he had many ingenious papers in- 

 serted in the Philos. Trans, from the 46th to the 60th volume. Mr. R. was a person of very honour- 

 able character and conduct, being greatly respected by the more learned and best characters among 

 the members of the Royal Society; on most occasions his opinions in the council were much regarded- 

 and he had the honour to be one of the committee chosen to inspect and report on the government's 

 powder magazine at Purfleet, concerning its damage and security from lightning. In his mode of 

 teaching, and arranging the materials in his publications, he was remarkably neat and methodical • a 

 habit which he probably in some measure acquired in imitation of his good friend and master Wm 

 Jones, Esq. many of whose papers, on his demise, passed into the hands of Mr. Robertson which 

 were afterwards sold by auction, along with the valuable library of the latter, after his death. 



VOL. X. N 



