† 54: 4:0 N. Inches ; required to find the Content of the greatest Conic-Fruftum that can be inscrib'd therein, in Wine Gallons. ( 10.).Q.0 ESTION the 52d, by R. F. ; Semi-Circle HG I be described, and let there be any other right Line A B, bifected in C, and from C erect a Perpendicular CD=GF; and the Perpendicular LN being erected upon any Point of the Diameter HI, let it be made as DC;AC:: HL:A 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. (11.) QUESTION the 53d, by J. T. ADMIT O Ports thus fituated be, d. One in the Latitude of * Thirty Three : * 33 : 0 N. d. Th' other in † Fifty Four North Latitude, Porty Degrees their difference of Longitude. d. Also in l Eighteen South an Iand lies, ll 18:0 $. Which from th' abovefaid Port, (I must premịse) Is Equi-diftant. Now I beg that you, How far the Ifland is from each, would shew. ( 12.) QUESTION the 54th, by J. T. L ET there be a Rhomboides, the two longer parallel Sides of which, are each equal to 82. Feet, and the two Shorter each equal to 55. Feer, and the acute Angle thereof 68 Degrees; its required to inscribe anEllipfis therein which Shall 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 Position and Magnitude of the Transverse and Conjugare Axes, and to construet the fame Geometrically, Having among some old Papers accidentally met with a small Pamphlet, called, The Mathematical Delights, printed at Newcastle, and as I cannot find by what En.. quiry. I have made, that there ever was any more than one published, and consequently no Answers to the Queffions therein contained : I have taken the Liberty to borrow the two following Ones from thence, and to B preleng " fail'd from to bear off present 'em to my Reader in the very famo Dress I tound them, (13.) QUESTION the 55th, by E. H. A , angled Triangle, the Fences of which being drove away by a Flood, and consequently thrown in common with the Neighbouring Ground, and having lost the Dimensions, only remember'd this, (viz.) that there food an Oak in the Neighbouring Ground, from which it a right Line were produced to each side of the Meas dow, in its neareit Diftance, would divide each side into iwo equal Parts ; the Lines being in Measure 81.25 318.75; and 458.95 Chains ; required the sides and Area of the faid Meadow. (14.) QUESTION the 56th, by R. €. TW WO Ships the Wiliam and Mary, fail from an Inand in North Latitude, and are bound to ad Inand in South Latitude; the Ship Mary, fails between the South and the West ; and the Ship William, fail hetween the South and the Eaft; they having made ata d. Angle of so": 37 between them, and then they find themselves upon the Equinoctial Line just 46 Leagues alunder the Mary Atill keeping the same Course as at first, Me falls in with the Mand. The William alters his Course from the Equino&tial, and runs 29 Leagues bei tween the South and the West, then he finds the INand the fame Course, falls in with the island where the Mary is : The Meridian Distance of this and is found to be 70 Leagucs Westward : from hence I demand both their Courfes steerd, and their Distance run, and the Latitude of the Iand failid from, as also the Latitude they are now in. Of Of the INVENTION of the METHOD of FLUXIONS. matical World made so great a Noife as the late Controversy between Sir Isaac 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 Analysi promote, only a very few Copies of it were printed and sent to the Fellows of the Royal Society, and to such Mathematicians as could judge of those Matters, nor are to be bought; but, fince that Time, it was translated out of Englith into Larin, and published in the Year 1722 ; from which Edition I shall select such 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 small Treatise, calla, A General Met bod of the quadrature of Curves by injis nite Series, communicated the fame to Mr. 7. Collins, herein is laid down a Demonstration of the Rule for the Quadrature of Curves, which is the common Foundation both of the Do&rine 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, Mt. Leibnitz was in England, and again in O Etober 1676, and the Interval of this Time, he spent in France, during wķich he kept a Correspondence with Mr. Oldenburgh, and by his Means, with Mr. Collins ; and sometimes, also 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 Division, after the Method of Mercator, to reduce the Fraction into a Series. Furthermore its evident, that Mr. Leibnitz did not then understand these Series , because in the same Letter, he defires that Mr. Newton would explain to him the Manner how he got those Series : Again, in a second Letter from Mr. Newton to Mr. Leibnits, dated O&tober 24, 1676, he gives yet clearer hints of his Method, and illustrates it" by Examples, and lays down a Rule · by which, from the Ordinates of certain Curves being given, their Arcas may be obtain'd in finite Terms, when it is poflible. By these Lights, and aflifted by these Examples, an ordinary Genius might have 'un, derstood the Newtonian Method, much less can we expect that it should yet lye conceald from the acute Mr. Leibnity: After this, viz. in the Year 1677 Mr. Oldenburgb dying, an End was put to this Correspondence, and Mr. Leibnitz returning to Hanover, he first published in the Ada Eruditorum Lipsienfia, for 1684. the Elements of the differential Calculus, nor do there appear the least 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 Isaac Newton's Books of the Number of Curves of the first Kind, and of the Quadrature of Figures, the Editors of the Leipfick AEts began to infinuate, that Mr. Leibnitz was the firft Inventor of the differential Calculus, and that Sir Isaac Newton had subftituted Fluxions for Differences, which are nearly as the Increments of Plowing Quantities generated in very small Particles of Time ; and had used them very elegantly, as well in his Mathematical Principles of Natural Philosophy, as in other Treatises, afterwards published by him. Just as Honoratus Faber in his Synopsis Geometrica, had fubftituted a Progreffion of Motion for Cavallerius's Method of Indivisibles. The Meaning of which Words, is this, that Mr. Neruton had substituted Fluxions for the Leibnitian Differences, as Honoratus Faber had substituted a Progression of Motion, for, The Method of Cavallexius : That is, that Ma Leibnits Leibnitz was the first Inventor of this Method, and that Mr. Newton had received the fame from him, and had substituted Fluxions for Differences. Excited by these Infinuations, Mr. J. KRIL, in a Let. ter printed in the Philosophical Transactions for the Months of September and October, 1708. wrote on the contrary, That, without all Dispute, Mr. NEWTON was she 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 same was afterwards published by Mr. LEIBNITZ in the Aeta Eruditorum, he having first changed the Name, and Manner of Notation. Mr. LEIBNITZ. being offended thereat, complained to the Royal Society against Mr. KEIL; and infilted, that he should publickly confefs his Fault, Dr. KEIL chose rather to answer in Writing to those Things of which LEIBNITZ had complained, in which Letters he more fully explained and vindicated what he had before asserted : But Mr. LEIBNITZ, not at all satisfy'd herewith, wrote again to the Royal Society, in which he stillcomplain'd of Dr. Keil, calling him an Upstart, and one that had little Knowledge of Matters tranfa&cd so long before, nor having a Commission from him who was chiefly interested in the Dispute, ( that is, from Sir Isaac NEW TON) and referr'd it to the Judge ment of the Society, whether such vain Bablings, and unjust Calumniations, ought not to be restrained. Mr. LEIBNITZ was in England in the Beginning of the Year 1673, and again in the Month of Otober 1676, and in that in terval fpent his Time in France, during which Space he had a Correspondence 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 published by Dr. WALLIS. Mr. NEWTON can not be a Witness on the Part of Mr. KEIL; nor Mr. LEIBNITZ on his own : But other Witness there is none remaining alive. Wherefore the RoyalSociety being ewise appealed to by Mr. LEIBNITZ against |