other informations, and shall request you to continue them as you have occasion.

Lately, in discourse with Mr. Halton, he was pleased to shew me a straight-lined projection for finding the hour by inspection, the sun's declination and height being given; but concealing the proportion from which it was derived, gave me occasion to vary it into this form (plate 5), of which I need say no more to you than that it is derived from this known proportion, demonstrable from the analemma:

As the difference of the sines of the sun's meridional altitude, and his altitude at 6 is to the radius or 6 hours so the difference of the sines of the said altitude at 6, and the altitude given to the sine of the hour from six.

By this sinical projection, which I have drawn with a radius of ten inches, I can find the hour of the day to a minute, except when the sun or star is near the meridian; and therefore I find it of great use to me, when I have not time to make use of my pen, to calculate the hour. It were easy to add the ecliptic to it, and parallels of latitude for five or six degrees on each side, which I intend to draw in some other-coloured ink in a large projection of this sort, of twenty inches radius, when the holidays are over. My success in this has caused me to set my proportions for finding the parallaxes in altitude, longitude, and latitude on such scales as they would permit, by which, if the planets be in the ecliptic, I can find them to half-a-minute's exactness by bare inspection. If they have latitude it is more troublesome, but I hope to facilitate it much, of which at better leisure you may hear more from, Sir, your affectionate friend and servant,

JOHN FLAMSTEED. The early part of this letter is in the Gen. Dict. vol. V. p. 258.

CXCII.

Sir,

FLAMSTEED TO COLLINS.

The Observatory in Greenwich, May 25, 1677.

I here send you the account of the experiments I promised you, which were made with the steel bow. It requires more time than I have to spare to delineate all the parts of the machine for discharging the bullet from it, and I suppose would not be very much desired by your friend. I shall run them over, in short, and I hope to your satisfaction. The place on which the shots were made was a level before the observatory, twelve score yards long or better, which has at each end a descent; on the western end of which a frame was fixed in the earth, so as the bow, being laid parallel to the horizon, might be also even with the plane of the earth: and on this frame was a piece of wood hung on a centre. To this a toothed arch was fastened, by the help of which, and a worm screw, the piece of wood which was to receive the bow, might be raised or depressed easily and conveniently, and thereby itself set to any inclination. The bow was of three good steel laths; the string, two thirds of an inch thick, had a small wooden plug fastened by one end to the middle of it; this was about a foot long or more, and had the other end somewhat bigger, almost as wide as the hollow of the cylinder, from which it cast the shot. This cylinder was about eighteen inches long, and so placed as that, when the string was drawn up, the end of the plug which served to discharge the shot might fall within it about one inch. The string was forced up

by the help of a long worm screw; and to take the inclinations of each shot there was provided a cylinder of wood, some seven or eight inches longer than the hollow brass one, which it was so turned as to fill, when it was put into it. To this a quadrant of seven inches was fastened, with a thread and plummet for taking the inclinations. At the first trials the wind was very little, or rather a perfect calm; at the latter it blew sometimes briskly, which was the reason, as was thought by gunners, why all the upper ranges fell short of the like made but four days before; and I cannot but allow of their opinion, for all the under ranges were defended from the wind by the hill, the upper, flying above it, were not.

The length of every shot was taken by a chain of twenty-two yards, or sixty-six feet, divided into an hundred links; the bullets were of brass, nearly all of the same bigness, and eleven ounces weight; but if any was heavier than another, that shot at the same range always fell nearer the bow.

They conclude the longest shot to be made at 42

degrees, and what they find by the shot from the bow, they find also in the arrow shot from it, and in the bullet discharged from the mortar-piece. The longest

range in all being about 42 degrees, and the rest of the ranges in one proportional to the like at the same elevations in the other.

I hope I have discharged my promise: if in any thing I may further inform you, any note shall command the services of

your friend and servant,

JOHN FLAMSTEED.

I doubt not you will remember your promise not to let any one know from whom you had these informations.

J. F.

CORRESPONDENCE WITH JAMES GREGORY.

CXCIII.

COLLINS TO GREGORY.

Worthy sir,

Mr. Thompson lent me to peruse one of your books, De quadratura circuli et hyperbolæ, and hath sent the other to Dr. Wallis. The book doubtless deserves much applause, and consequently the press. It is the first I have seen of that argument, illustrated in numbers. At the end of a French grammar, I am informed there is a catalogue of mathematical books, amongst which it is said there is one of this title, L'Egalité de la parabole avec un ligne droit par M. Fermat and if so, as Van Heurat at the end of second edition of Descartes, the hyperbola is squared by necessary consequence. The quadrature of the hyperbola was promised by the Lord Brounker, as in Dr. Wallis his first epistle to L. B., also by Dr. Wallis

himself, page 51 Commercii Epistolici, but about it I have formerly received what follows from Mr. Barrow of Cambridge.

*

and I have some papers of Mr. Warner deceased, wherein he proves if parallels be drawn to one asymptote, so as to divide the other into equal parts, the spaces between them, the hyperbola, and asymptote, are in musical progression, the which, if desired, I may communicate. The quadrature of the hyperbola is a proposition very necessary in gauging, and consequently of great use in relation to the king's revenue; for many brewers' tuns are like silver tankards, trunci conici circulares, divided into two partitions with a plane erect to the base to hold liquors of different strengths, and also stand stooping, and the quadrature of the hyperbola doth capacitate us to cube any segment of a cone. Let us take such a one as the plane passing through the base doth cut an hyperbola, through the same chord line in the base let another plane pass, so as likewise to pass through the vertex of the cone, then by this latter plane is the cone divided in such ratio as the segments of its base; and the wedge between

A

D

V

P

B

both planes is of the conic figure whose base is the hyperbola, whose height is a perpendicular falling from the vertex of the cone on the hyperbolic plane: and so in any case where the plane passeth through the base, the proposition being of frequent use. One end of my writing is to excite you to render it ticable; to this purpose I send you some approaches

easy and prac