THE BRITISH CRITIC, For JANUARY, 1794. QUI MONET QUASI ADJUVAT. PLAUT. ART. I. Tables of Logarithms of all Numbers, from 1 to 101,000; and of the Sines and Tangents to every Second of the Quadrant. By Michael Taylor, Author of the Sexagefimal Table. With a Preface and Precepts for the Explanation and Ufe of the fame, by Nevil Mafkelyne, F. R. S. Aftronomer Royal. Very large quarto. 41. 4s. Wingrave. EVERY publication which tends to the abridgment of la bour, and the promotion of accuracy, must be acceptable to the literary world, but more particularly fo to that part of it for whose use the work was intended. If we measure life by the multiplicity or importance of the things tranfacted, we may confider performances of this nature as contributing to its prolongation; for they enable us to investigate fubjects of fuch extent and variety, as without affiftance we should be unable to examine. Among devices of the above-mentioned tendency, and among contrivances for affifting man in his approximations to truth, logarithms hold a very distinguished place. They aid the mathematician, in many of the higher departments of abstract science, in carrying on his demonftrations, in freeing his reafoning from complex and cumberous expreffions, and in enabling him to exhibit his conclufions in the most neat and elegant terms. Recourfe is had to them in every enquiry to which trigonometry, either plane or spherical, can be applied. WheBRIT, CRIT. VOL. III. JAN. 1794. B ther ther our efforts be directed to the calculation of the heights. and distances of objects on land, the failing of a veffel on the ocean, or the motions of the planets in the vast expanse of the heavens, we turn to logarithms for a mitigation of our labour, and with full confidence that they will bring us, not only to a quick, but also to an accurate conclufion. Logarithms have been defined a fet of artificial numbers, whofe relation to the natural numbers, or thofe in common use, is fuch, that their addition and fubtraction answer to the multiplication and divifion of the natural numbers of which they are the logarithms. Thus the logarithm of 5 being added to the logarithm of 4, the fum will be the logarithm of 20, the natural number produced by multiplying 5 by 4; and, on the contrary, the logarithm of 5 being fubtracted from the logarithm of 20, the remainder will be the logarithm of 4, the natural number arifing from the divifion of 20 by 5. And hence it is evident, that computations in which the multiplication and divifion of high natural numbers would be neceffary, must be much expedited by means of logarithms, thefe artificial numbers being first calculated and arranged in tables, in columns oppofite to the natural numbers to which they respectively belong. The calculator, provided with fuch affiftance, turns to the proper pages of his tables, and having felected the particular logarithms, fit for his immediate purpose, advances with eafe to his wifhed-for determination, instead of going through a fatiguing operation, and of course incurring a great hazard of making mistakes. The volume, in which Baron Neper, the inventor, announced this curious and highly ufeful difcovery to the world, is a small quarto, written in Latin, and is entitled Mirifici Logarithmorum Canonis defcriptio, ejufque ufus, in utraque Trigonometria, ut etiam in omni Logistica Mathematica, ampliffimi, facillimi, expeditiffimi, explicatio. It opens with a dedication to the Prince of Wales, afterwards Charles the Firft; and this is followed by a Preface, fetting forth the confiderations which acted as incitements to the invention, and making it known. to mathematicians. This hiftory of logarithms, after that time, has been executed by Dr. Hutton, of Woolwich, in a a manner which does honour to his induftry, learning, and abilities, in his publication entitled Mathematical Tables; containing common hyperbolic and logistic Logarithms, &c. In this work the reader will find a fuccinct account of trigonometrical tables, and a full and diftinct review of the feveral methods used for the conftruction of logarithms; and fhould he wish to peruse the methods at full length, in the words of their respective authors, the elegant collection of them by Francis Maferes, Efq. Cur Curfitor Baron of the Court of Exchequer, will afford him complete fatisfaction *. Mr. Taylor, the author of the tables now before us, did not live to fee their publication. He died before the five laft pages of the table of logarithmic fines and tangents were printed; and, in confequence of this event, the prefent Aftronomer Royal entered upon the remaining part of the labour. To Dr. Mafkelyne, therefore, we are indebted not only for the fuperintendance of the above-mentioned parts of the table, but also for the preface, and the whole of the matter prefixed to the logarithmic tables, occupying 64 pages fully printed. These parts appear to us to be worthy of their refpectable author, and to be executed with that zeal, learning, and ability, with which he has ever espoused and supported the caufe of fcience. We lay the following part of the Preface before our readers, as it contains the reafons for a new publication on the fubject, and a statement of Mr. Taylor's intentions, labours, and anxious endeavours at correctnefs. "We are indebted to Napier, Briggs, and Vlacq, for their ingenious inventions and induftrious labours, in providing us with our prefent logarithmic tables, as to the fubftance; fome improvements in the form and difpofition of them only, having been introduced by later authors. Gardiner's Tables of Logarithms, of Numbers, and Logarithms of Sines and Tangents, for every ten feconds of the quadrant, which are the most complete tables published fince Vlacq's, are confeffedly taken from Vlacq's, only abridged to 7 places of decimals. The trouble of making proportion for the intermediate feconds, or computing equations of 2d and 3d differences, &c. where the differences are very irregular, has alone been complained of." "To obviate thefe difficulties, the late Mr. M. Taylor undertook this laborious work of computing the logarithmic fines and tangents to every fecond of the quadrant, by interpolating Vlacq's logarithmic fines and tangents, whereby he obtained a table to every fecond, confifting of ten decimal places of figures, as Vlacq's did, which he then abridged to only 7 decimal places befides the Index, taking particular care to make the last figure true to the nearest unit over or under, a circumftance that will be found very conducive to exactness in fuch cafes, where an unit in the laft place is of confequence, and where feveral logarithms are added together." In the collection here mentioned, feveral explanatory tracts, compofed by Mr. Baron Maferes, are inferted. The two first volumes of the work, beautifully printed in quarto, were published in 1791, and from the Preface we have reafon to hope for a third. "Nor did the author ufe lefs care and diligence in fupervifing the prefs and correcting its errors. He generally examined 3, and fometimes 4 fucceffive proofs, with the help of an affiftant, one reading while the other hearkened. The first proof he compared with his manufcript, attending chiefly to the 2, 3, or 4 laft figures, according as the differences rendered it neceflary; and further examined the index, and 3 or 4 or 5 first decimal figures himself fingly. He alfo compared the 2d proof, as to the 2, 3, or 4 laft figures; and again as to the two laft, with the manufcript; then at every 36" with Brigg's Trigonometria Britannica; and, laftly, to every 10" with Vlacq's and Gardiner's Tables. He further took the differences of the two laft figures of the fucceffive numbers by inspection, He examined the tangents and their correfponding co-tangents, by trying whether their fum every where came out 10; and, if any difference appeared, detected where the error lay, whether in the tangents or co-tangents, by the differences. By this method, and with this care, the 3d proof was generally rendered correct to the author's mind; if not, another, and another were taken till the prefs, was found correct, and the sheet was then worked off. The table of logarithms of num bers was compared in the printing, with the beft tables of lo garithms, and particularly with thofe of Dr. Hutton's Mathematical Tables. "The five laft pages of the table of logarithmic fines and tangents were all that remained unfinished at the prefs when the author died. Thefe I examined carefully myself in the fame manner as the author had the reft of the work: only I examined the tangents and co-tangents, by trying whether the fum of the correfponding figures in every place, beginning from the left, made 9, except the last fignificant figures to the right, which fhould make 10, which comes to the fame thing with the author's method." The pages which intervene between the Preface and Mr. Taylor's part of the volume, contain the explanation and examples of the ufe of the logarithmic tables. Under this general title, the nature of these artificial numbers is first confidered; and then very full inftructions are given for their management in calculation. Logarithmic arithmetic, in the abstract, fucceeds thefe precepts; and immediately after this we are presented with a variety of rules and problems, illuf-. trating the foregoing doctrine, In many of these our learned author has not been content with a bare application of logarithms, but has fubmitted to the view of the reader fuch matter as is closely connected with the examples under confideration. To the ftudent this must be acceptable, as it prefents him with information; to the experienced it is convenient, as it recals to his mind what he had formerly inveftigated, and confequently enables him to proceed without doubt or hefi tation. The following is an enumeration of the articles adduced for illuftrating the use of logarithms, or for explaining the nature of the examples. The rule of proportion.-Rule for the combination and arbitration of exchange.-Compound intereft.-General properties of plane triangles.-The folutions of the cafes of right-angled plane triangles.-The solutions of the cafes of oblique plane triangles.-General properties of fpherical triangles.-General properties of right-angled spheric triangles.-Table of folutions of the cafes of right-angled fpheric triangles, with remarks.-Solutions of oblique fpheric triangles.-General properties of a rectilateral fpheric triangle, or having one fide equal to 90°.-The folutions of a rectilateral fpheric triangle, with remarks.-Table of accurate folutions of cafes of right-angled fpheric triangles by logarithms, with remarks.-Table of accurate folutions of oblique fpheric triangles by logarithms, with remarks.-Table of accurate folutions of a rectilateral fpheric triangle, by logarithms, with remarks. Such of the above articles as relate to spherical trigonometry, have more particularly engaged our attention. We find in them fome new improvements, and feveral original remarks, which, being obferved, will guard the calculator against errors, and guide him to accurate refults. The remaining pages, previous to the logarithmic tables, contain fifteen curious and useful problems, of which the following fhort but clear account is given in the Preface: "The fecond and third of thefe, which are, to find the logarithm of the fum, or difference of two numbers, whofe logarithms are given, are taken from Mr. Cagnoli's ufeful treatife of plane and fpheric trigonometry. The following problems, from the third to the eighth, are folved according to the fame principles. The eighth and ninth problems are fubfervient to the folution of the eleventh problem, or of cubic equations. The ninth contains the folution of a famous problem, that of finding any power of an impoffible binomial, in terms of another impoffible binomial, which I have derived from the analogy of the circle to the equilateral hyperbola. There is fomething of this kind propofed and partly executed by Mr. de Moivre, at the end of Sanderfon's algebra. His extraction of the cubic root of an impoffible binomial is juft and complete, but not fo fimple as what is here given. His rule for extracting any root out of any given power of an impoffible binomial, gives B 3 rightly |