Inches; required to find the Content of the greatest Conic-Fruftum that can be infcrib'd therein, in Wing Gallons.

UP

(10.) QUESTION the 52d, by R. F.

PON the right Line HI (vide Fig. 15.) let the Semi-Circle HGI be defcribed, and let there be any other right Line A B, bifected in C, and from C erect a Perpendicular CDGF; and the Perpendicular LN being erected upon any Point of the Diameter HI, let it be made as DC; AC HLA P, and in P let the Perpendicular P M be erected NL: It's required to find the Locus of all the Points M, found after this Manner.

(IL) QUESTION the 53d, by J. T.
d.
ADMIT two Ports thus fituated be,
One in the Latitude of * Thirty Three:

Th' other in † Fifty Four North Latitude,
Forty Degrees their difference of Longitude.
Alfo in Eighteen South an Ifland lies,

33:0 N.

d.

54:0 N.

d.

18:0 S.

Which from th' abovefaid Port, (I must premife)
Is Equi-diftant. Now I beg that you,

How far the Ifland is from each, would fhew.

L

(12.) QUESTION the 54th, by J. T. ET there be a Rhomboides, the two longer parallel Sides of which, are each equal to 82. Feet, and the two fhorter each equal to 55. Feet, and the acute Angle thereof 68 Degrees; its required to infcribe an Ellipfis therein which fhall touch the four Sides: Or what amounts to the fame Thing, given the two conjugate Diameters of an Ellipfis: 82. and 55. Feet, and the Angle they contain 68 Degrees; to find the Pofition and Magnitude of the Tranfverfe and Conjugate Axes, and to conftruct the fame Geometrically.

Having among fome old Papers accidentally met with a fmall Pamphlet, called, The Mathematical Delights, printed at Newcastle, and as I cannot find by what Enquiry, I have made, that there ever was any more than one published, and confequently no Answers to the Queffions therein contained: I have taken the Liberty to borrow the two following Ones from thence, and to prefens

B 2

prefent 'em to my Reader in the very fame Drefs I found them,

A

(13.) QUESTION the 55th, by E. H.

Gentleman had a Meadow, in Form an Oblique angled Triangle, the Fences of which being drove away by a Flood, and confequently thrown in common with the Neighbouring Ground; and having loft the Dimenfions, only remember'd this, (viz. that there food an Oak in the Neighbouring Ground, from which if a right Line were produced to each Side of the Mea dow, in its nearest Diftance, would divide each Side into two equal Parts; the Lines being in Measure 81.25; 318.75; and 458.15 Chains; required the Sides and Area of the faid Meadow.

(14.) QUESTION the 56th, by R. C.

TW WO Ships the William and Mary, fail from an Ifland in North Latitude, and are bound to an Iland in South Latitude; the Ship Mary, fails between the South and the Weft; and the Ship William, fails he tween the South and the Eaft; they having made an

d.

Angle of 50: 37 between them, and then they find themselves upon the Equinoctial Line juft 46 Leagues áfunder ; the Mary till keeping the fame Courfe as at first, fhe falls in with the Illand. The William alters his Course from the Equinoctia), and runs 20 Leagues be tween the South and the Weft, then he finds the Inland fail'd from to bear off him due North, he ftill keeping the fame Courfe, falls in with the inland where the Mary is: The Meridian Distance of this Ifland is found to be 70 Leagues Weftward: from hence I demand both their Courfes fteer'd, and their Distance run, and the Latitude of the Inland fail'd from, as alfo the Latitude they are now in.

Of

Of the INVENTION of the METHOD of FLUXIONS.

THERE has not perhaps any Thing in the Mathematical World made fo great a Noife as the late Controverfy between Sir Ifaac Newton and Mr. Leibnitz, concerning the Invention of the Method of Fluxions, (called by Foreigners, the Differential Calculus) the whole State of the Cafe was published in the Year 1715, in a Book call'd, Commercium Epiftolicum Collinii et aliorum de Analyfi promota, only a very few Copies of it were printed and fent to the Fellows of the Royal Society, and to fuch Mathematicians as could judge of thofe Matters, nor are to be bought; but, fince that Time, it was tranflated out of English into Latin, and publifhed in the Year 1722; from which Edition I fhall felect fuch Paffages as may give the Reader a clear View of the Dispute, and enable him to judge thereupon.

In the Year 1669, Dr. Barrow, having received from Sir Ifaac (then Mr.) Newton, a fmall Treatife, call'd, A General Method of the Quadrature of Curves by infinite Series, communicated the fame to Mr. F. Collins, herein is laid down a Demonftration of the Rule for the Quadrature of Curves, which is the common Foundation both of the Doctrine of Fluxions, and of the differential Calculus, by which its evident he had invented the Method before that Time. In the beginning of the Year 1673, Mr. Leibnitz was in England, and again in October 1676, and the Interval of this Time, he spent in France, during which he kept a Correfpondence with Mr. Oldenburgh, and by his Means, with Mr. Collins ; and fometimes, alfo with Mr. Newton, from the last of whom, he received a Letter dated 13th of June, 1676, wherein is taught the Method of reducing Quantities into infinite Series, that is, of exhibiting the Increments of flowing Quantities: This Method was utterly unknown to Mr. Leibnitz, before he received the abovefaid Letter of Mr. Newton's, as he himself acknowledges in his Letter to Mr. Oldenburgh, dated August 27, 1676, for before

that

that Time, he was obliged to transform an irrational Quantity into a rational Fraction, and then by Divifion, after the Method of Mercator, to reduce the Fraction into a Series. Furthermore its evident, that Mr. Leibnitz did not then understand thefe Series,because in the fame Letter, he defires that Mr. Newton would explain to him the Manner how he got thofe Series: Again, in a second Letter from Mr. Newton to Mr. Leibnits, dated October 24, 1676, he gives yet clearer hints of his Method, and illuftrates it by Examples, and lays down a Rule by which, from the Ordinates of certain Curves being given, their Areas may be obtain'd in finite Terms, when it is poffible. By thefe Lights, and affifted by these Examples, an ordinary Genius might have un derstood the Newtonian Method, much lefs can we expect that it should yet lye conceal'd from the acute Mr. Leibnitz: After this, viz. in the Year 1677 Mr. Oldenburgh dying, an End was put to this Correfpondence, and Mr. Leibnitz returning to Hanover, he first published in the Ata Eruditorum Lipfienfia, for 1684 the Elements of the differential Calculus, nor do there appear the leaft Footsteps, (nor does he himself pretend) that he had the Method be fore the Year 1677, that is, after he had received the two above-mentioned Letters from Mr. Newton. But upon the Publishing of Sir Ifaac Newton's Books of the Number of Curves of the firft Kind, and of the Quadrature of Figures, the Editors of the Leipfick Ats began to infinuate, that Mr. Leibnitz was the firft Inventor of the differential Calculus, and that Sir Ifaac Newton had fubftituted Fluxions for Differences, which are nearly as the Increments of Flowing Quantities generated in very small Particles of Time; and had ufed them very elegantly, as well in his Mathematical Principles of Natural Philofophy, as in other Treatifes, afterwards publifhed by him. Juft as Honoratus Faber in his Synopfis Geometrica, had fubftituted a Progreffion of Motion for Cavallerius's Method of Indivifibles.

The Meaning of which Words, is this, that Mr. Newton had fubftituted Fluxions for the Leibnitian Differences, as Honoratus Faber had fubftituted a Progreffion of Motion, for, The Method of Cavallerius: That is, that Mr

Leibnitz

Leibnitz was the firft Inventor of this Method, and that Mr. Newton had received the fame from him, and had fubftituted Fluxions for Differences.

Excited by thefe Infinuations, Mr. J. KEIL, in a Let ter printed in the Philofophical Transactions for the Months of September and October, 1708. wrote on the contrary, That, without all Difpute, Mr. NEWTON was the firft Inventor of the Algorithm of Fluxions, as was evident to any one, that had read his Letters, published by Dr. Wallis; and that the fame was afterwards publifhed by Mr. LEIBNITZ in the Acta Eruditorum, he having firft changed the Name, and Manner of Notation. Mr. LEIBNITZ, being offended thereat, complained to the Royal Society against Mr. KEIL; and infifted, that he should publickly confefs his Fault. Dr. KEIL chofe rather to anfwer in Writing to thofe Things of which LEIBNITZ had complained, in which Letters he more fully explained and vindicated what he had before afferted: But Mr. LEIBNITZ, not at all fatisfy'd herewith, wrote again to the Royal Society, in which he ftill complain'd of Dr. KEIL, calling him an Upftart, and one that had little Knowledge of Matters tranfacted fo long before, nor having a Commiffion from him who was chiefly interested in the Difpute, (that is, from Sir ISAAC NEW TON) and referr'd it to the Judgment of the Society, whether fuch vain Bablings, and unjuft Calumniations, ought not to be restrained. Mr. LEIBNITZ was in England in the Beginning of the Year 1673, and again in the Month of October 1676; and in that Interval fpent his Time in France, during which Space he had a Correfpondence with Mr. OLDENBURGH, and by his Mediation with Mr. COLLINS and Mr. NEWTON: NOW upon what he learnt in England, or from that Intercourse of Letters before-mention'd, all the Question turns. Mr. OLDENBURGH and Mr. COLLINS are long ago dead: As for Mr. NEWTON, he then refided at Cambridge, and knew little more of the Matter than what appear'd from his own Letters afterwards publifhed by Dr. WALLIS. Mr. NEWTON can not be a Witnefs on the Part of Mr. KEIL; nor Mr. LEIBNITZ on his own: But other Witness there is none remaining alive. Wherefore the Royal Society being twice appealed to by Mr. LEIBNITZ against