An Account of a Book

VIII. An Account of a Book, intituled, Harmonia Mensurarum, sive Analysis & Synthesis per Rationum & Angulorum mensuras promotae: accedunt alia Opuscula Mathematica: per Rogerum Cotesium. Edidit & auxit Robertus Smith, Coll. Trin. Cantab. & Reg. Soc. Socius; Astronomiae & Experimentalis Philosoph. post Cotesium Professor. Cantabrigiae 1722. in 4to. Prostant apud Bibliopolas Londinenses. The Book consists of three Parts. In the first, called Logometria, the Author's chief Design is to shew how that sort of Problems, which are usually reduced to the Quadrature of the Hyperbola and Ellipsis, may be reduced to the Measures of Ratio's and Angles; and afterwards be solved more readily by the Canons of Logarithms and Sines and Tangents. He defines the Measures of Ratio's to be quantities of any kind, whose Magnitudes are analogous to the Magnitudes of the Ratio's to be measured. In this Sense any Canon of Logarithms is a System of numerical Measures of the Ratio's of the absolute Numbers to an Unit: The Parts of the Asymptote of the Logistic Line, intercepted between its Ordinates, are a System of Linear Measures of the Ratio's of those Ordinates: The Areas of an Hyperbola, intercepted between its Ordinates to the Asymptote, are a System of Plane Measures of the Ratio's of those Ordinates: And since there may be infinite Systems of Measures according as as various kinds of Quantities are made use of, such as Numbers, Time, Velocity, and the like; or according as the Measures of any one System may be all increased or diminished in any given proportion; in such Variety much Confusion may possibly arise as to the Kind and absolute Magnitudes of particular Measures, which happen to fall under Consideration. Our Author very happily removes this Difficulty; by shewing that the Nature of the Subject points out the Measure of a certain immutable Ratio for a Modulus in all Systems, whereby to determine the Kind and absolute Magnitudes of all other Measures in each System. The first Proposition is to find the Measure of any proposed Ratio. This he considers in a way so simple and general, as naturally leads to the Notion and Definition of a Modulus; namely, that it is an invariable Quantity in each System, which bears the same Proportion to the Increment of the Measure of any proposed Ratio, as the increasing Term of the Ratio bears to its own Increment. He then shews, that the Measure of any given Ratio is as the Modulus of the System, from whence it is taken: and that the Modulus in every System is always equal to the Measure of a certain determinate and immutable Ratio, which he therefore calls the Ratio Modularis. He shews that this Ratio is expressed by these Numbers $2,7182818 \ldots$ to $1$, or by $1$ to $0.3678794 \ldots$. So that in Briggs's Canon the Logarithm of this Ratio is the Modulus of that System: In the Logistic Line the given Subtangent is the Modulus of that System: In the Hyperbola the given Parallelogram, contained by an Ordinate to the Asymptote and the Absciss from the Center, is the Modulus of that System: and in other Systems the Modulus is generally some remarkable Quantity. In the second Proposition he gives a concise concise uncommon Method for calculating Briggs's Canon of Logarithms; together with Rules for finding intermediate Logarithms and Numbers, even beyond the Limits of the Canon. In the 3d Proposition he constructs any System of Measures by a Canon of Logarithms; not only when the Measure of some one Ratio is given, but also without that Datum, by seeking the Modulus of the System by the Rule abovementioned. In the 4th, 5th, and 6th Propositions he squares the Hyperbola, describes the Logistic Line and Æquangular Spiral by a Canon of Logarithms, and shews some curious Uses of these Propositions in their Scholia. Take an easy Example of the Logometrical Method, in the common Problem for finding the Density of the Atmosphære. Supposing Gravity uniform, every one knows, that if Altitudes are taken in any Arithmetical Progression, the Densities of the Air in those Altitudes will be in a Geometrical Progression; that is, the Altitudes are the Measures of the Ratio's of the Densities below and in those Altitudes, and so the difference of any two Altitudes is the Measure of the Ratio of the Densities in those Altitudes. Now to determine the absolute or real Magnitude of these Measures, the Author shews, a priori, that the Modulus of the System is the Altitude of the Atmosphære, when reduced everywhere to the same Density as below. The Modulus therefore is given (as bearing the same Proportion to the Altitude of the Mercury in the Barometer, as the specific Gravity of Mercury does to the specific Gravity of Air) and consequently the whole System is given. For since in all Systems the Measures of the same Ratio's are analogous among themselves; the Logarithm of the Ratio of the Air's Density in any two Altitudes will be to the Modulus of the Canon, (that is, to the Logarithm of A a the *Ratio Modularis* defined above,) as the difference of those Altitudes is to the aforesaid given Altitude of the homogeneous Atmosphere. He concludes the Logometria with a *General Scholium*, containing great Variety of elegant Constructions both Logometrical and Trigonometrical; such as give the Length of Curves either Geometrical or Mechanical; their Area's and Centers of Gravity; the Solids generated from them, and the Surfaces of these Solids; together with several curious Problems in Natural Philosophy, concerning the Attraction of Bodies, the Density and Resistance of Fluids, and the Trajectories of Planets. Several of these Problems have two Cases; one constructed by the Measure of a Ratio, and the other by the Measure of an Angle. The great Affinity and beautiful Harmony of the Measures in these Cases, has given occasion to the Title of the Book. The Measures of Angles are defined (just as the Measures of Ratio's) to be Quantities of any Kind, whose Magnitudes are analogous to the Magnitudes of the Angles. Such may be the Arcs or Sectors of any Circle, or any other Quantities of Time, Velocity, or Resistance, analogous to the Magnitudes of the Angles. Every System of these Measures has likewise its *Modulus* homogeneous to the Measures in that System, and may be computed by the Trigonometrical Canon of Sines and Tangents, just as the Measures of Ratio's by the Canon of Logarithms; for the given *Modulus* in each System bears the same Proportion to the Measure of any given Angle, as the *Radius* of a Circle bears to an Arc which subtends that Angle, or the same as this constant Number of Degrees $57,2957795130$ bears to the Number of Degrees in the said Angle. Upon the whole our Author thus expresses himself, p. 35. "Ex adductis adductis hactenus exemplis, Geometris integrum erit de methodo nostrâ judicare; quam quidem, si pro- ba fuerit, ulterius excolere pergent & excolendo la- tius promovebunt. Patet utique campus amplissi- mus in quo vires suas experiri poterunt, praesertim si Logometriae Trigonometriam insuper adjungant, quibus miram quandam affinitatem in se invicem euntibus intercedere notabam. Hisce quidem prin- cipiis haud facile crediderim generaliora dari posse; cum tota Mathesis vix quicquam in universo suo am- bitu complectatur praeter Angulorum & Rationum Theoriam. Neque sane commodiora sperabit, qui animadverterit effectiovis facilitatem per amplissi- mas illas, omnibusque suis numeris absolutas, tum Logarithmorum, tum Sinuum & Tangentium tabu- las; quas antecessorum nostrorum laudatissimae solertiae debemus acceptas. Ut vero tanti beneficii uberior nobis exsurgat fructus, id nunc exponendum restat, quibus artibus ad istius modi conclusiones re- ctissimâ perveniatur. In hunc finem Theoremata quædam tum Logometrica tum Trigonometrica ad- jecisset, quæ parata ad usum asservo; ni consul- tius visum esset, quum absque nimiis ambagibus ea tradi non possent, intacta potius præterire atque aliis denuo investiganda relinquere. Why the Author takes his Principles to be so gene- ral, will farther appear by an Instance or two. In the Problem already mentioned he measures the Ratio of the Air's Densities in any Altitudes, by the Altitudes themselves, making use of the Altitude of an uniform Atmosphere for the Modulus. So likewise when he considers the Velocities acquired, and the Spaces de- scribed in given Times, by a Body projected upwards or downwards in a resisting Medium with any given Velocity; he shews, that the Times of Descent, added to a given Time, are the Measures of Ratio's, to a given Modulus of Time, whose Terms are the Sum and Difference of the ultimate Velocity and the present Velocities that are acquired: that the Times of Ascent, taken from a given Time, are the Measures of Angles, to a given Modulus of Time, whose Radius is to their Tangents in the Ratio of the ultimate Velocity to the present Velocities: and lastly, that the Spaces described in Descent or Ascent, are the Measures of Ratio's to a given Modulus of Space, whose Terms are the absolute accelerating and retarding Forces arising from Gravity and Resistance taken together at the Beginning and End of those Spaces. This general Account may suffice to illustrate what I am going to say; that since the Magnitudes of Ratio's (as well as their Terms) may be expounded by Quantities of any Kind, the Mathematician is at Liberty upon all Occasions to choose those which are fittest for his Purpose; and such are they without doubt, that are put into his Hand by the Conditions of the Problem. He may indeed represent these Quantities by an Hyperbola, or any other Logometrical System, were not his Purpose answer'd with greater Simplicity by the very System itself, which occurs in each particular Problem. And the same may be said for the Systems of Angular Measures, instead of recurring upon all Occasions to Elliptical or Circular Area's. As to the Convenience of calculating from our Author's Constructions, he shews that the Measures of any Ratio's or Angles are always computed in the same uniform Way; by taking from the Tables the Logarithm of the Ratio, or the Number of Degrees in the Angle, and then by finding a fourth Proportional to three given Quantities; for that will be the Measure required required. The simplest Hyperbolic Area may indeed be squared by the same Operation taught in the Author's fourth Proposition; but the simplest Elliptic Area requires somewhat more: Those that are more complex in both Kinds (which generally happens) require an additional Trouble to reduce them to the simplest: to square them by infinite Series is still more operose, and does not answer the End of Geometry. Upon the whole therefore it may deserve to be considered, for what Purposes should Problems be always constructed by Conic Areas, unless it be to please or assist the Imagination. The Design of Theoretical Geometry differs from Problematical; the former consists in the Discovery and Contemplation of the Properties and Relations of Figures for the sake of naked Truth; but the Design of the latter is to do something proposed, and is best executed by the least Apparatus of the former. The Logometria was first published by the Author himself, in the Philosoph. Transact. of the Year 1714. No 338. But his Logometrical and Trigonometrical Theorems abovementioned were not published till after his Decease. These Theorems make the second Part of the Book, and are calculated to give the Fluents of Fluxions (reduced to 18 Forms) by Measures of Ratio's and Angles; in such a manner, that any Person may perfectly comprehend their Construction and Use, though altogether unacquainted with Curvilinear Figures, as expressed by Equations. And this Circumstance does also render the Application of them to the Analysis and Construction of Problems extremely easy. Of this kind the Author has given a great many choice Examples both in abstract and physical Problems; which make up the third and last Part of the Book. The Author, a little before his Decease has informed us (in a Letter of May 5. wrote to his Friend Mr. Jones) "that Geometers had not yet promoted the inverse Method of Fluxions, by Conic Areas, or by Measures of Ratio's and Angles, so far as it is capable of being promoted by those Methods. There is an infinite Field (says he) still reserved, which it has been my Fortune to find an Entrance into. Not to keep you longer in Suspense, I have found out a general and beautiful Method by Measures of Ratio's and Angles for the Fluent of any Quantity which can come under this Form $$\frac{d^{\theta_n + \delta_n - 1}}{e + fz^n},$$ in which $d, e, f$ are any constant Quantities, $z$ the variable, $n$ any Index, $\theta$ any whole Number affirmative or negative, $\delta$ any Fraction whatever. The Fluents of this Form which have hitherto been considered are $$\frac{d^{\theta_n - 1}}{e + fz^n} \quad \text{&} \quad \frac{d^{\theta_n + \delta_n - 1}}{e + fz^n}:$$ These you remember are Sir Isaac Newton's two first, and from these all his others are easily deduced. And as his irrational Forms of the quadratrick Kind are derived from the rational, so from my general rational Form I deduce irrational ones of all Kinds. For instance, if $\delta$ represent any affirmative or negative Fraction, the Fluent of any Quantity of this Form $$d^{\theta_n - 1} \times \frac{e + fz^n}{g + hz^n},$$ or of this $d^{\theta_n - 1}$ $$\times \frac{e + fz^n}{g + hz^n},$$ and so of some others, depends upon the "the Measures of Ratio's and Angles. Mr. Leibnitz in the Leipzig Acts of 1702, p. 218 and 219, has very rashly undertaken to demonstrate, that the Fluent of \( \frac{x}{x^4 + a^4} \) cannot be expressed by Measures of Ratio's and Angles; and he swaggers upon the Occasion (according to his usual Vanity) as having by this Demonstration determined a Question of the greatest Moment. Then he goes on thus; as the Fluent of \( \frac{x}{x + a} \) depends upon the Measure of a Ratio, and the Fluent of \( \frac{x}{x^2 + a^2} \) upon the Measure of an Angle; so he had more than once expressed his Wishes, that the Progression may be continued, and it be determined to what Problem the Fluents of \( \frac{x}{x^4 + a^4}, \frac{x}{x^8 + a^8}, \text{&c.} \) may be referred. His Desire is answered in my general Solution, which contains an infinite Number of such Progressions. I can go yet farther, and shew him how by Measures of Ratio's and Angles, without any Exception or Limitation, the Fluent of this general Quantity \[ \frac{dz}{e + fz^n + gz^{2n}} = \frac{\theta n + \delta}{\lambda n - 1} \] or even this \[ \frac{dz}{e + fz^n + gz^{2n} + hz^{3n}} = \frac{\theta n + \delta}{\lambda n - 1} \] may be had; where \( \theta \), as before, represents any Integer, and the Denominator \( \lambda \) of the Fraction \( \frac{\delta}{\lambda} \), presents any Number in this Series, 2. 4. 8. 16. 32. &c. any whole Number being denoted by its Numerator \( \delta \). In truth I am inclined to believe, that Mr. Leibnitz's grand Question ought to be determined the contrary Way; and that it will be found at last, that the Fluent of any rational Fluxion whatever, does depend upon the Measures of Ratio's and Angles, excepting those which may be had in finite Terms even without introducing Measures. Dr. Taylor knowing by this Letter what the Au- thor had done, was pleased to propose the Invention of the Fluents of the two last Fluxions as a Problem to the Mathematicians in foreign Parts. Mr. Ber- noulli in the Leipsic Acts of 1719. p. 256, did shew accordingly how they are reducible to Conic Area's. The Editor has published the Author's own Solution by Measures of Ratio's and Angles; and upon this Foundation has constructed new Tables of Log- ometrical and Trigonometrical Theorems, for the Fluents of Fluxions reduced to 94 Forms, part rational and part irrational. He has likewise added general Notes upon the chief Difficulties in the Book, together with a Me- thod of composing Synthetic Demonstrations of Lo- gometrical and Trigonometrical Constructions, illu- strated by various Examples. The first Treatise in the Miscellaneous Works is concerning the Estimation of Errors in Mixt Mathe- matics. It consists of 28 Theorems, to determine the Proportions among the least contemporary Variations of the Sides and Angles of Plane and Spherical Tri- angles, while any two of them remain invariable. An Example will shew their great Use in Astronomy. The Time of the Day or Night is frequently to be deter- mined by the Altitude of some Star. Let it then be proposed to find the Error, that may arise in the Time, from any given Error in taking the Altitude. By ap- plying the 22d Theorem to the Triangle form'd by the Complements of the Star's Altitude and Declination and by the Complement of the Pole's Elevation, the Author shews, that the Variation of the Angle at the Pole, and consequently the Error in Time, will be as the Error in the Altitude directly, as the Sine Complement of the Pole's Elevation inversely, and as the Sine of the Star's Azimuth from the Meridian inversely. Consequently, if the Error in the Altitude be given, under a given Elevation of the Pole, the Error in Time will be reciprocally as the Sine of the Azimuth contained by the Meridian and the Vertical which the Star is in. This Error therefore will be the same, whatever be the Altitude of the Star in the same Vertical; and will be least when the Vertical is at right Angles to the Meridian. But will be absolutely the least in the same Circumstance, if the Observer be under the Æquator. In which Case, if the Error in the Altitude be one Minute, the Error in the Time will be four Seconds. If the Observer recedes from the Æquator towards either Pole, the Error will be increased in the Proportion of the Radius to the Sine Complement of the Latitude: So that in the Latitude of 45 Degrees it will be $5\frac{1}{2}$ Seconds, and in the Latitudes of 50 and 55 it will be $6\frac{3}{4}$ and $6\frac{7}{8}$ Seconds respectively. If the Star be in any other vertical Oblique to the Meridian, the Error will still be increased in the Proportion of the Radius to the Sine of that oblique Angle. Lastly, if the Error in the Altitude be either bigger or less than one Minute, the Error in Time will be bigger or less in the same Proportion. Much after the same manner may the Limits of Errors be computed in other Cases, which arise from the Inaccuracy of Observations, and from hence the most convenient Opportunities for observing are also determined. The Second Treatise is concerning the Differential Method. The Author having wrote it, before he had seen Sir Isaac Newton's Treatise upon that Subject, has handled it after a manner somewhat different. The Title of the Third Treatise is Canonotechnia or concerning the Construction of Tables by Differences. It consists of ten Propositions, most admirably contrived for expeditious Computation of intermediate Terms in any given Series. The last Proposition, which contains a general Solution of the whole Design, is this; Datis seriei cujuscunque terminis aliquot equidistantibus, quorum intervalla secunda sunt in aquales quotlibetcunq; partes, propositum sit invenire terminos interserendos. The Book concludes with three small Tracts, concerning the Descent of Bodies, the Motion of Pendulums in the Cycloid, and the Motion of Projectiles, composed in a very natural and easy manner. The Author has wrote some other Pieces, yet unpublish'd, which the Editor has given an Account of in his Preface to the Book. The Reader will find every Subject treated with uncommon Elegance and Simplicity. FINIS. LONDON: Printed for W. and J. Innys, Printers to the Royal Society, at the Prince's Arms the West End of St. Paul's Churchyard.