Графични страници
PDF файл
ePub
[ocr errors]

hearers, I fear thefe honourable diftinctions are not their due. And what will be their doom, if they live and die without repenting of their finful, their cruel, their unchriftian conduct? Hear the word of God: "They fhall have judgment without mercy, who have fhewed no mercy." P. 294.

[ocr errors]

Neither in this, however, nor in the other inftances that follow it, does Dr. Booker appear to take a judicious method of illuftrating the fubject; and indeed, on the whole, we thould be inclined to caution him to have, in future, a greater fear of the prefs; did we not conclude, from the lift of names prefixed, that there were, poffibly, fome ftrong reafons for the prefent publication. We will endeavour therefore to find a better fpecimen. In doing which, we find that we have marked the following paffage of Sermon V. as containing one of the moft ori. ginal and judicious fuggestions in the volume:

"Yet I would not have you fuppofe that fympathizing with the unfortunate, is a more convincing proof of a good heart than partici pating the joy of the happy. That perfon who will do the latter will very feldom fail to do the former; while the reverfe is by no means true. There is nothing to be envied in mifery: and envy is a pernicious alloy to all greatnefs of foul. Many are envied in profperity by those who would have wept with them in adverfity. Such perfons weep more from a morbid conflitutional love of melancholy, than from a regard for the person with whom they apparently sympathize. In particular, let me caution you againft allowing yourfelves credit for fenfibility of heart, on account of the tears which you fhed at beholding fictitious fcenes of forrow. Many have their eyes fuffufed with tears, on these occafions, who can behold real mifery with unconcern: nay, who can be the cause of mifery, and plunder and opprefs the Orphan and the widow." P. 70.

To each fermon, except two, is fubjoined an appropriate prayer, taken chiefly from the collects and other parts of the liturgy, with additions and correclions by the author; who has here executed his tafk very well. But no author, not even the venerable Dr. Johnfon, will be wronged, if Mr. Nelfon fhould be pronounced to remain unequalled in this way. See his morning and evening prayers for a family, at the end of his companion for the festivals and fafts of the church of England." The name of Nelfon is juftly dear to chriftians: and the numerous impreflions of his book afford a comfortable hope, that piety is not fo faft declining amongst us as we are fometimes, perhaps infidiously, taught to believe.

[ocr errors]

ART

ART. XII. Torelli's Archimedes

[Continued from No. III. page 325.]

ADvancing in the volume before us, the treatife on the sphere

and cylinder next meets our eye. It confifts of two books, and has been esteemed by the most able mathematicians of the laft and prefent century, as one of the greatest inftances of human penetration. The illuftrious author appears to have been convinced that this was his principal work, as he desired, according to Plutarch, in his Life of Marcellus, that a sphere and cylinder might, after his death, be reprefented upon his monument. This feems to have been religiously obeyed. Cicero, when quæftor of Sicily, vifited Syracufe, and, with the most lively veneration for the memory of the departed philofopher of the place, fought for his tomb. He found it, as he informs us, in the fifth book of his Tufculan Questions, furrounded with brambles and bushes; but obferved on a little column a delineation of the above-mentioned figures, and, the place being cleared, he perceived fome mutilated verses.

Almoft the whole of the epiftle to Dofitheus, prefixed to the first book on the fphere and cylinder, is wanting in the Bafil edition, but in the prefent it is complete. From this we learn, that Archimedes had previoufly fent him the quadrature of the parabola, and that Eudoxus first discovered that a pyramid is the third part of a prism, and a cone the third part of a cylinder, the bafes and altitudes being equal. The epiftle is immediately followed by fuch truths as he thought proper to affume as first principles; and these are fucceeded by feveral propofitions relating to the infcription and circumfcription of figures in and about a circle, cone, and cylinder, which pave the way to the higher parts of the treatise.

The great object of Archimedes, in this book, was to determine the ratio between the furface of a sphere, and that of its circumfcribing cylinder, and alfo the ratio between the folids themfelves. In order to obtain thefe, he demonftrates that the convex furface of a right cylinder is equal to a circle, whofe radius is a mean proportional between the fide of the cylinder and the diameter of its base; and that the convex furface of an ifofceles cone is equal to a circle, whofe radius is a mean proportional between the fide of the cone, and the radius of the bafe: hence it eafily follows, that the convex surface of an ifofceles cone is to its base as the fide of the cone to the radius of the bafe. This opens to him the way, not only of eftimating the convex furface of the lower fruftum of an ifofceles cone con

tained bteween two planes parallel to the bafe, but also that of afcertaining the folidity of conical rhombs; parts of them between planes parallel to the common bafe of the cones of which they are compofed, and parts of them between the furfaces of isosceles cones. The propofitions in which thefe figures are confidered, and two concerning equilateral polygons of an even number of fides infcribed in a circle, form a complete preparation for the objects in view: the remaining ftep was to connect them with the sphere. For this purpose Archimedes supposes an equilateral polygon, whofe number of fides is a multiple of four, to be infcribed in a great circle of the sphere, and the circle and polygon to move about a fixed diameter joining two of the oppofite angles of the polygon. In confequence of this motion the circle generates the fphere, and the polygon a folid. inscribed in it; and this infcribed folid is made up of figures already examined. For at the two angular points, joined by the diameter about which the circle and polygon revolved, there are two ifofceles cones, and if planes parallel to their bases be paffed through the other angular points, the remaining part of the infcribed figure will be divided into fuch frufta of cones as we have already mentioned. A fimilar polygon is circumfcribed about a great circle of the fphere, and a like rotatory motion being underflood, a folid fimilar to the infcribed is circumfcribed about the fphere. By means of thefe figures, he demonstrates that the furface of a sphere is equal to the quadruple of one of its great circles; and that the fphere itself is equal to the quadruple of a cone, whofe bafe is a great circle, and altitude the radius of the fphere. From hence he infers, that a sphere being infcribed in a clylinder, the whole furface of the cylinder is fefquialter of that of the fphere, and the cylinder itfelf fefquialter of the fphere. The fame method alfo enabled him to determine that the convex furface of a fegment of a sphere is equal to a circle, whofe radius is equal to a straight line drawn from the vertex of the fegment to the circumference of its bafe; and that a fpherical fector is equal to a cone, whose base is equal to the fpherical furface of the fector, and whofe altitude is equal to the radius of the fphere. In the course of his advancement to these important determinations, many other properties are demonftrated, curious in themfelves, and far removed from common obfervation.

The fecond book on the sphere and cylinder confifts of seven problems and three theorems. In the problems it is proposed, -to find a fphere equal to a cone or cylinder ;-to cut a given fphere, fo that the convex furfaces of the fegments, and the fegments themselves may have a given ratio to one another ;-to find a fegment fimilar to one and equal to another fegment, or

5

fimilar

fimilar to one and having its spherical furface equal to that of another;—from a given sphere to cut a fegrnent which shall have a given ratio to a cone, having the fame bafe and altitude as the fegment. The theorems refpect the fegments of a fphere;their ratio to cones connected with them;-the limits of their ratio to one another;-and in the last it is demonstrated, that of all fegments under equal spherical furfaces, a hemifphere is the greatett.

This fecond book either did not come fo highly finished from the hand of the author as the preceding, or it has fuffered by the ignorance and careleffness of tranfcribers. The latter we are moft inclined to believe, as the demonftrations in it consist chiefly of long compofitions, refolutions, and contorfions of ratios, and therefore uncommon care was neceffary to prevent the eye from being mifled by the frequent repetitions of the fame words. To whatever cause the lofs is to be attributed, the reader has to regret the omiffion of several important steps in the reafoning of this book.

The measure of the circle (circuli dimenfio) which stands next in the volume, is one of the most important articles in geometry; and, without doubt, was confidered by Archimedes as neceffary to the completion of his treatife on the sphere and cylinder: for the practical eftimation of the magnitudes of the furfaces and folids, there confidered, ultimately depends upon that of the circle. It confifts of three propofitions; in the first of which, by the infcription and circumfcription of polygons, he proves that a circle is equal to a right angled triangle, having one of the fides about the right angle equal the radius of the circle, and the other fide round the right angle equal to the circumference. In the fecond propofition he proves that the ratio of the circle to its circumfcribed fquare is nearly as 11 to 14; but the truth affumed in this, is that upon which it ultimately refts, and it is not demonftrated before the third or laft propofition. For in this fecond, he fuppofes the circle to be equal to a right angled triangle of which one fide round the right angle is equal to the radius and the other equal to 22 of

7

it. In his 3d propofition he proceeds to his approximation towards the ratio between the diameter of a circle, and its circumference. This is founded upon the equality of a fide of an equilateral hexagon, infcribed in a cirele to the radius; by means of which, and the 3d prop. of B. 6. and the 47th prop. of B. 1. of Euclid, he approximates to the ratio of the fide of an equilateral polygon of 96 fides, circumfcribed about, and infcribed in a circle, to the diameter; and from thence concludes, if the diameter of a circle, be 1, the circumference will be lefs:

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

two laft expreffions, in decimals, is 3*1428571, &c. the latter is 3*1408450, &c. and the diameter being 1, according to the more accurate approximations of the moderns, the circumference is 3.1415926, &c. Several attempted the measure of the circle before Archimedes, but none with the fame fuccefs which attended his labours.

All the treatises of which we have given an account are accompanied, excepting the quadrature of the parabola, with Commentaries of Eutocius; but his obfervations on the measure of the circle conftitute the end of his remarks on Archimedes, to the great regret of the attentive reader.

Eutocius was born at Afcalon in Palestine, and flourished about the middle of the fixth century. His Commentaries on the Conics of Apollonius he addreffed to Anthemius; from what we have of his in the prefent volume, we learn that Ifidorus was his preceptor; and, according to Procopius, Anthemius and Ifidorus were the two architects of the church of Saint Sophia, built at Conftantinople about the year 532..

Eutocius very feldom paffes over a difficult paffage in his author without explaining it, or a chafm in the reafoning without fupplying the defect. His remarks are ufually full; and fo anxious is he to render the text perfpicuous, that fometimes he enters upon elucidation where, in our opinion, Archimedes is fufficiently clear. As he does not wander from the subject matter before him, our readers may form a general idea of the nature of his explanations, from what we have faid of the text : but his Commentaries on the fecond book of the Sphere and Cylinder deserve more particular notice. As in this part his exertions were more neceffary, fo they are more frequent, and fometimes various methods are offered for fupplying a deficiency. The most remarkable instance of this kind originates in the fecond propofition, where Archimedes fuppofes the method of finding two mean proportionals between two given straight lines to be understood, and therefore paffes over the manner of obtaining them in filence. From hence Eutocius takes occafion to introduce, at full length, the methods employed by Plato, Hero, Philo Byzantinus, Apollonius, Diocles, Pappus, Sporus, Menechmus, Architas, Eratofthenes, and Nichomedes, to folve this curious and useful propofition. These endeavours, it seems, arofe from a defire to effect that famous problem among the ancients, the duplication of the cube; the only difficulty in the folution of which confifts in the finding of two mean proportionals. The methods of the above mentioned mathematicians are most of them ingenious in theory; but we think that of NiGg chomedes BRIT. CRIT. VOL. 1. AUG. 1793.

« ПредишнаНапред »