the computation of it in the common way, by computing the value of its initial terms, abfolutely impracticable. For Sir Ifaac Newton has obferved concerning this feries in that extreme cafe (when then becomes r- + 3 5 r 7 &c.) and another feries that is almoft as flow as this, that to exhibit its value exact to twenty decimal places of figures, there would be occafion for no less than five thousand millions of its terms, to compute which would take up above a thousand years. Now in this extreme cafe Mr. Maferes has fhown that to bring out the numbers true to the third decimal place, 500 of the terms of the original feries must be computed, and that the fame degree of exactnefs will be attained by computing only eight terms of the differential feries, to which he has reduced the original one. But the beft method, he fays, of applying the differential feries to the investigation of the value of one of these very flow feriefes, is to compute a moderate number of the first terms of the flow feries in the common way, and then apply the dif ferential feries to the computation of its remaining terms. He fhews the advantage of this hod of proceeding by applying it to the foregoing example.. cafe of an arch of 45°, which by the help of eight terms only of his differential feries is brought out rx. 785, 398, 165, &c. which is true to the eighth place, the more exact value being rx.785, 398, 163, &c. which degree of exactnefs could not have been attained by the mere computation of the original feries, without computing fifty millions of its terms. The author proceeds to give another example in computing of the feries, which expreffes the time of the defcent of a pendulum through the arch of a circle: concluding his paper, as follows. "I have endeavoured to find another differential feries, fimilar to that above defcribed, for the purpose of investigating the value of an infinite feries of this form, to wit, a + bx + cxx + dx3 + ex2 + ƒx3 +8x® + bx+ &c. (in which all the terms are marked with the fign+, or are added to the first term a) when the co-efficients b, c, d, e, f, g, h, &c. decrease very flowly, and x is very nearly equal to 1, and the terms of the feries decreafe confequently fo flowly as to make the fummation of it in the common way, or by the mere computation and addition of its terms, almost impracticable; but my endea vours have not been attended with fuccefs. I may therefore, from my own experience, fubfcribe to the truth of what is afferted upon this fubject by the very learned and ingenious Mr. James Stirling in his Treatife, intitled, Summatio Scrierum, p. 17. to wit, that Series quarum termini funt per vices ucgativi et affirmativi, funt magis tractabiles quam Hh 2 altera, alteræ, ubi de Summatione agitur; though at first fight one would be apt to imagine the reverfe of this propofition to be true." Art. XII. is a Tranflation into Latin of a Passage in Ebn Younes; with fome Remarks thereon: in a Letter from the Rev. George Coftard, M. A. Vicar of Twickenham, to the Rev. Samuel Horfley, LL. D. Sec. R. S. This paffage relates to the obfervations and calculations of certain eclipfes of the fun and moon in ancient times. Art. XII. Obfervations on the Annual Evaporation at Liverpool in Lancashire; and on Evaporation confidered as a Teft of the Moisture or Drynefs of the Atmosphere. By Dr. Dobson of Liverpool. Communicated by John Fothergill, M. D. F. R. S. Art. XIV. An Account of Perfons who could not diftinguifh Colours. By Mr. Jofeph Huddart, in a Letter to the Rev. Jofeph Priestley, LL.D. F. R. S. This account of a peculiarity of vision in three brothers, is fingular and curious. The perfon, from whom it was taken, lived at Maryport in Cumberland. "His name, fays the writer, was HARRIS, by trade a shoe-maker. I had often heard from others that he could difcern the form and magnitude of all objects very diftinctly, but could not diftinguish colours. This report having excited my curiofity, I converfed with him frequently on the fubject. The account he gave was this: That he had reafon to believe other perfons faw fomething in objects which he could not fee; that their language feemed to mark qualities with confidence and precifion, which he could only guefs at with helitation, and frequently with error. His firft fufpicion of this arofe when he was about four years old. Having by accident found in the street a child's stocking, he carried it to a neighbouring houfe to inquire for the owner: he obferved the people called it a red stocking, though he did not understand why they gave it that denomination, as he himself thought it completely defcribed by being called a flocking. The circumftance, however, remained in his memory, and together with subsequent obfervations led him to the knowledge of his defect. As the idea of colours is among the first that enters the mind, it may perhaps seem extraordinary that he did not obferve his want of it ftill earlier. This, however, may in fome measure be accounted for from the circumstance of his fami y being quakers, among whom a general uniformity of colours is known to prevail. "He obferved alfo that, when young, other children could difcern cherries on a tree by fome pretended difference of colour, though he could only distinguish them from the leaves by their difference of size and shape. He obferved alfo, that by means of this difference of colour they could fee the cherries at a greater diftance than he could, though he could fee other objects at as great a diftance as they; that is, where the fight was not affifted by the colour. Large objects he could fee as well as other perfons; and even the fmaller ones if they were were not not enveloped in other things, as in the cafe of cherries among the leaves. I believe he could never do more than guefs the name of any colour; yet he could distinguish white from black, or black from any light or bright colour. Dove or ftraw-colour he called white, and different colours he frequently called by the fame name: yet he could difcern a difference between them when placed together. In general, colours of an equal degree of brightness, however they might otherwife differ, he frequently confounded together. Yet a striped ribbon he could diftinguish from a plain one; but he could not tell what the colours were with any tolerable exactnefs. Dark colours in general he often mistook for black, but never imagined white to be a dark colour, nor a dark to be a white colour. He was an intelligent man, and very defirous of understanding the nature of light and colours; for which end he had attended a course of kctures in natural philofophy. He had two brothers in the fame circumstances as to fight; and two other brothers and fifters who, as well as their parents, had nothing of this defect. One of the first mentioned brothers, who is now living, is master of a trading veffel belonging to Mary-port. I met with him in December 1776, at Dublin, and took the opportunity of converfing with him. I wished to try his capacity to diftinguish the colours in a prifm, but not having one by me, I asked him. Whether he had ever feen a rain-bow? He replied, He had often, and could diftinguish the different colours; meaning only, that it was composed of different colours, for he could not tell what they were. I then procured and fhewed him a piece of ribbon': he immediately, without any difficulty, pronounced it a ftriped and not a plain ribbon. He then attempted to name the different ftripes: the several stripes of white he uniformly, and without hesitation, called white: the four black stripes he was deceived in, for three of them he thought brown, though they were exactly of the fame fhade with the other, which he properly called black. He fpoke, however, with diffidence as to all thole tripes; and it must be owned, the black was not very distinct: the light green he called yellow; but he was not very pofitive: he faid, "I think this is what you call yellow." The middle stripe, which had a flight tinge of red, he called a fort of blue. But he was moft of all deceived by the orange colour; of this he spoke very confidently, faying, "This is the colour of grafs; this is green.' "I alfo fhewed him a great variety of ribbons, the colour of which he fometimes named rightly, and fometimes as differently as poffible from the true colours. I asked him, Whether he imagined it poffible for all the various colours he faw, to be mere difference of light and fhade; whether he thought they could be various degrees between white and black; and that all colours could be compofed of these two mixtures only? With fome hesitation he replied, No, he did imagine there was fome other difference. Art. Art. XV. A new Theory of the Rotatory Motion of Bodies affected by Forces difturbing fuch Motion. By Mr. John Landen, F. R, S. This is a new, and very ingenious Theory; but we cannot pretend to give our Readers any adequate, idea thereof, without the diagrams, or by any extracts. We fhall therefore content ourselves with recommending it to the careful perufal of all able Mathematicians. The Author ftands in no need of our commendations. He has long ago fo diftinguished himself as to rank with the Eulers and Bernoullis, &c. of the Age. Art. XVI. Directions for making the beft Compofition for the Metals of reflecting Telescopes; together with a Defcription of the Procefs for grinding, polishing, and giving the great Speculum the true parabolic Curve. By Mr. John Mudge; communicated by Alexander Aubert, Efq. F. R. S. This paper contains a moft excellent practical treatife, on the conftruction of reflecting telescopes; well worth the attentive perufal of ingenious artifts. The remaining articles are Art. XVII. Extract of a Register of the Barometer, Thermometer, and Rain, at Lyndon, in Rutland, 1776. By Thomas Barker, Efquire. Communicated by Sir John Pringle, Bart. P. R. S. Art. XVIII. Extract of a Meteorological Journal for the Year 1776, kept at Briftol, by Samuel Farr, M. D. Art. XIX. Meteorological Journal kept at the House of the Royal Society, by Order of the Prefident and Council. We take leave of this publication, by recommending to the editor a greater attention to the numbering of the articles; which, in this, as in fome preceding parts, is very incorrect: neither agreeing with their order of fucceffion, nor with the table of contents. A Tract on the Law of Nature, and Principles of Action in Man, By Granville Sharp. 8vo. 4s. White. From the exordium, of this elaborate and multifarious tract, we were led to conceive the reputable author intended to confider the Law of Nature in a political light, and apply his obfervations on the fubject to prefent times and circumftances. "I have neither leifure nor abilities, fays he, to undertake a regular definition of the Law of Nature, with all the doctrines ufually 5 ranked ranked under that head and indeed, if I had both leifure and abilities, I should want inclination; because fuch a work would unavoidably become voluminous, on account of the variety of authors neceffary to be mentioned, who have treated the fubject with different views: and as all science is vain, which is not reduced to practice, fo the more voluminous any fubject is rendered, the lefs it can be ufeful, on account of the increafed difficulty of communicating it to the generality of readers. I have therefore confined my tract to fuch general remarks on the fubject, as are most neceffary for the obferva-, tion of my countrymen at large, with refpect (more particularly) to one point, viz, the Illegality of reducing or fubjecting mankind to invo-. luntary fervitude, either under political or private dominion as all pretenfions to an unlimited authority of any man or men over others, are contrary to Natural Equity and the Laws of God, as well as baneful to mankind in general; which effect is unhappily demonstrated by the numberless inftances of unnatural oppreffion now prevailing to the deftruction of mankind, in almost every part of the world. "The Law of Nature, continues he, has been variously represented; but all the best writers, both ancient and modern, agree in adopting that maxim of the Civil Inftitutes, which declares involuntary fervitude, or flavery, to be " contrary to the Law of Nature: this rule is commonly understood as applicable only to domeflie flavery; but it is equally true when applied to political oppreffion, or the exercife of an unlimited dominion over a whole nation. Some few authors indeed have been fo unreasonable, as to affert that "there is no fuch thing as natural Law;" but they are properly cenfured by the learned Baron Pufendorf, in his "Law of Nature and Nations." Book 2. Chap. 3. He particularly mentions the argument of Carniades as contracted by Lactantius to the following effect. "That men firft inftituted Laws to fecure and promote their own advantage, &c. but that there was no fuch thing as Natural Law in the world," &c. p. 104. "Such doctrine is certainly convenient for Tyrants and Slaveholders of every degree, who mutt otherwife remain without excufe, whenever "the Law of Nature," and the Common Rights of Humanity," are urged against them: it is therefore neceflary for them, either to mifreprefent the Law of Nature (as the Reverend Mr. Thompson has done), or elfe (like Carniades) utterly to deny its exiflence. This latter method has been also adopted by fome modern advocates for Slavery, who, in private difcourfe on this fubject, bave declared, that they efteem "the Law of Nature" to be no other than their natural propenfity to pursue their own heart's defire of profit or pleafure: and this they call natural Liberty;" though it certainly is the most unnatural Tyranny: for when the immutable neceffity of reciprocal confide ration is forgot, or fet afide, there can be no fafety among men, and confequently no natural Liberty: we mult, therefore, fubmit ourfelves to be the fervants of law, in order to be truly free; according to the Author of a defence of the African Slave Trade, to which our author made a public reply. excellent |