private affairs, being out of the farthing office, or public concern, on account of tin farthings that are to ensue.

Frenicle's Triangles, Dary's Interest, and De la Hire's Conics, be pleased to accept, and to return Pascal du T[riangle] A[rithmetique], it being none of mine. The French Algebra is common to be had at 12s. price. I do not find he takes notice of two late Dutch authors, namely, Kinkhuysen and Abraham de Graaf, each of which I much more esteem than the French author. As to your tract of Combinations, I looked upon it as worthy public view, but conceiving it not so proper to go alone as with your other works, as thinking it might not prove of common sale, and being loth it should be printed at a house not used to mathematical work, nor furnished with proper types, you mistake if you think I had any design to hinder its coming forth on any other account, saving only that I thought I might give you some hints further to enlarge the subject of figurate arithmetic with the following advancements.

1. Leibnitz asserts all equations relating to figurate numbers or angular sections may have their roots expressed in surds, and hath imparted something of that kind by what he hath already attained, and promised further to explain himself when he arrives at Hanover, being here in October on his return to the Duke of that name, under whom he will be in eminent employ.

2. There may be combinatory tables made in species, which may be of as much use in Algebraic affairs, as the table of sines in practical geometry, in which tables most difficult problems shall be found ready solved, or easily thence deducible, and hence canons derived for the surd roots of all equations. His notions about the same he offered to explain on condition any one would undertake the work.

3. How to add or interpole progressions of squares, cubes, or of any other ranks of numbers, whose last differences are equal, is already performed by aid of tables of figurate numbers, by Mercator, in his Logarithmotechnia, and by Dary, in his Gauging.

4. The said Mercator continues a table of logarithms thereby.

5. The tables of logarithms may be carried on, and such progressions added, on supposal of the use of such figurate numbers; but by quite evading the same, either without or by fitting equations to all such ranks.

6. More particularly if you can add the reciprocals of an arithmetical progression, or its squares, by aid of equations suited thereto, derived from the knowledge of figurate arithmetic, you know as much, even in this kind, as I have hitherto attained. But lest you should not apprehend my meaning, I have set down two such ranks :

Where you see in the first rank, if the numerator be common, and the denominators in arithmetical progression, it is called the reciprocals of the arithmetical progression; the latter rank is called the reciprocals of the squares of an arithmetical progression. For the finding the present worth of an annuity of 100 per annum, to continue 100 years, discounting 6 per cent. per annum simple interest, I have added 100 terms in the following rank; viz.

and found the sum of them to amount to (as I remem

ber) 32007. Os. 1d. fere, which argues the absurdity of simple interest in relation to purchases. The explication of the method is not only scientific but large.

Mr. Broome hath your Treatise of Combinations, and is resolved to go forward with it, which I shall as much promote as in me lieth: these accessions may be proper enlargements for a following edition.

I remain

This letter is dated 1676 in the original MS. It must, however, be 1676-7, although he speaks of not having written since August. The fact may have escaped his memory, and the date of some of the books mentioned, as well as of Leibnitz' visit to London, must be considered as determining the point.

CORRESPONDENCE WITH WALLIS.

CCXCI.

COLLINS TO WALLIS.

Jan. 2, 1665.

Reverend Sir,

The unexpected favour I received from you when at Oxford, cannot but make me mindful of those obligations of thanks I ought to return, as a testimony whereof vouchsafe to accept of a few books I herewith send, viz.

Mengoli Geometria Speciosa. Antimi Farbii Opusculum Geometricum. Caravagii Mediolanensis Applicationum Doctrina, which handles the limits of such equations as have but two nomes. Not that I think any man's works can add to your vast treasure of knowledge, but possibly may excite you to supply

what they have omitted, or amend what they have.but perfunctorily performed.

It hath pleased God still to reserve me, as a monument of his mercy, in the land of the living; but upon my return hither, I found wanting Mr. Anthony Thompson and Mr. Henry Sutton, two of the best mathematical instrument makers. And the quondam servant of Mr. Sutton, John Marke, being now returned, is about taking his master's house; he desires the presentment of his most humble service. We hope he may prove as good a workman as his deceased master, and that mite of knowledge I have attained to [I] shall most willingly serve him with.

Some new books are published in France; as Pascal du Triangle Arithmetique, &c. and some others, whereof I hope ere long to have an account, from the hands of an ingenious German gentleman now residing at Paris, and familiar, I presume, with M. Roberval, who, being on the contemplation of curves, perchance will have proposed to his consideration the curve in Wright's or Mercator's sea-chart, that represents the great arch of a circle, and with what plane such a curve may be cut out of the cylinder, in which Mr. Wright supposes the sphere inscribed; whether the said meridian line be not the same, or in the same ratio as Mr. Gunter's logarithm scale of tangents, beginning therein at 45 degrees, and counting each half degree a whole one; and the nature of that curve that may be made by raising the degrees of that meridian line as perpendiculars upon the degrees of the equinoctial in that chart, which are equal parts and the contrary; and this in order to the more easy description of the great arch's curve in that chart, or for the supply of the meridian line, it not yet being known (geometrically) to find the rhumb between two places

of known latitudes and given difference of longitude. And if any thing come to my knowledge about these methods, I should be willing to communicate it, and humbly crave the favour, when your occasions draw you to London, to afford me the cognizance thereof, that I may have some further opportunity administered of expressing my gratitude. In the interim, if you shall vouchsafe a line or two in return, it needs no other direction but to me, as an accountant at the Excise office in St. Bartholomew Lane, London, who desires to be accounted an admirer of your worth, and your most affectionate servitor

CCXCII.

COLLINS TO WALLIS.

Reverend Sir,

Feb. 28.

instant, with

I have received yours dated the your enclosed exercise, and considerations about Mercator's, or rather Mr. Wright's sea-chart, for the use whereof [I] return most hearty thanks, but withal am sorry I should put you to so much trouble about such a subject, which, I presume, I as far understood as those authors left it, if not farther. This must redound to my crime and your goodness, who were pleased to divert your studies from other sublime speculations. You have discovered where the difficulty lies, &c. may it redound to your immortal fame to remove it! not that I presume to desire [you to]

i The sentence in this place has been left incomplete by the erasures and interlineations introduced into it: it is therefore print

ed so as to express, as nearly as possible, what appeared to be the meaning it was intended to cORvey.