Plane Trigonometry becomes next in courfe the fubject of confideration: and in the definitions the author very properly obferves, that an Arc of 90° has not (according to the definition of titole terms) either a tangent or a fecant. He inflances the abfurdity of the fuppofition, that fuch an Arc has a tangent or fecant, from a right-angled fpherical triangle, where radius: cofine of the angle at the bafe :: tangent of the hypothenufe: tangent of the bafe; now when the bafe 90°, the hypothenufe 90°; and therefore thefe Arcs being equal, if they have any tangents, of whatever value they may be, they must be equal, and therefore radius cofine of the angle at the bafe, whatever that angle may be. This falfe conclufion arifes from the falfe fuppofition, that an Arc of 90° has a tangent. The author afterwards gives another inftance of a falfe conclufion arifing from the fame fuppofition.

In refpect to the propofitions, the author appears to have fe lected all that are likely to be useful in philofophical or any other enquiries. He has clearly pointed out the ambiguous cafes; and where there can be any poffible difficulty, he has fhown how the rules for computation are to be adapted to a logarithmic operation: and here he observes, that if an Arc be found in terms of its cofine, and the Arc be very finall, or near 180°, the variation of the cofine will be fo fmall, that it will not vary for many feconds. Thus, if the log. cofine came out 9.9999998*, then in the tables this is the cofine of an Arc from 2. 52" to 3.41"; here is therefore an Arc for 49", which has the fame cofine in the tables, owing to their being continued to feven decimals only; it is impoffible therefore to say what Arc from 2', 52" to 34 41", we are here to take. In fuch cafes, it is here obferved, that the expreffion must be transformed into one where the fine enters inftead of the cofine. In like manner, if an Arc be near 90°, and be expreffed by the fine, the expreffion must be changed into one where the cofine

enters.

The principles here delivered are next applied to find the heights of objects; to carry on a menfuration of a country by a feries of triangles; to find the length of an Arc of the me ridian, &c. after which, examples are given of computing all the different cafes of triangles, by Logarithms. To this is added an Appendix, fhowing how to find the powers of the fine and coune of an Arc; to conftru&t a table of fines, cofines, &c. to exprefs the fine and cotine of an Arc in terms of the

This by mistake is printed 9,999998. See p. 72, line 4th from the bottom.

3

impoffible

impoffible quantity, remarking their use in physical aftronomy; and to exprefs an Arc in terms of the fines of multiples of that Arc; and here the author shows, that if z be any Arc, z = fin. z } fin. 2 z + 1 fin. 32 —, &c. ad infinitum.

The author next proceeds to Spherical Trigonometry; and. here he begins with definitions, and the description of fuch circles of the sphere as the fubject neceffarily requires; and these, the reader will find, are explained, and their properties investigated, with more than ufual accuracy. He then goes on to the explanation and investigation of all the general properties of triangles, in which are found many things not ufually met with in works of this kind, but which are frequently found ufeful in removing the ambiguity which arifes in the solution of Spherical Triangles.

The folution of right-angled fpherical triangles, by Napier's circular parts, is the next fubject of confideration; and here, by the arrangement of the middle part, the adjacent extremes, and the oppofite extremes, in a table, directly against each other, for all the cafes, the whole is rendered extremely evident. The demonftration of the two Theorems, for all the cafes, is made very eafy by means of two propofitions, the proofs of which take up only a few lines. The equations for all the cafes are arranged in a table, fo as to correfpond to the other table. By this method, all the cafes are contained in a inuch fmaller compafs, and are much more easily remembered, than when they are refolved into fo many proportions as they neceffarily muft, which can never, without great labour, be committed to memory. The author has purfued the fame method for a quadrantal triangle; that is, a triangle having one fide 90°. which he has shown can be refolved by the circular parts; and he has arranged the equations in a table accordingly. The ambiguous cafes are pointed out; and fuch general properties of right-angled and quadrantal triangles are given, as mult very frequently tend to remove an ambiguity which might otherwife arife. The neceffity alfo of attending to the figns of the quantities made ufe of; that is, of the fines, cofines, tangents, &c. is fhown in feveral inftances. Some further proportions of right-angled triangles are added; and fome properties of oblique-angled triangles are demonftrated, from letting fall a perpendicular from the vertical angle upon the bafe. The folution of oblique-angled triangles is next given and here the reader will find all the cafes investigated in a clear and fimple manner, and the rules very plainly ftated; fome of which investigations are new: and to thefe are fubjoined a great many affections of spherical triangles, which will be

found

found extremely ufeful in removing ambiguities which frequently arife in the computations of fpherical triangles.

After having delivered every thing which can be useful in the theory, the author proceeds to the practice, fhowing how to compute the various cafes by Logarithms; and bere he has chofen fome examples in aftronomy; among which, he has given a direct folution of the following very ufeful problem : Given two obferved Altitudes of the Sun, and the Time between, with the Change of Declination in the Interval of the Obfervations, and the Declination at the firft; to find the Latitude of the Plane. An inveftigation of this is firft given, and then a rule is deduced for a logarithmic computation; to which, is added an example. The rules given for the folution of this problem have generally been partly by tentative methods, approximating to the truth; nor has the change of declination been. before confidered. The author therefore, by giving an eafy practical rule for a complete folution, has done an important Tervice to the navigator, to whom this is principally of use. At lard, the two obfervations may be made at the fame place; but on board a ship in motion, the obfervations will be made in different places; in this cafe, the altitude taken at the fecond obfervation must be reduced to what it would have been, if the observation had been made at the place where the first altitude was taken; for the method of doing which, the author refers to his Political Aftronomy; a work, which contains a very full defcription of the construction and use of all aftronomical inftruments, in their lateft ftate of improvement.

The variation of fpherical triangles is the next, and last subject of this Treatife. Cotes was, we believe, the first perfon who wrote any thing on this fubject; it was published in the Harmonia Menfurarum, under the title of De eftimatione Errorum in mixta Matheft. The author has here first confidered the variation of right-angled fpherical triangles, in which fome new properties are given, one of which we conceive may be. very frequently ufeful; that is, if the angle at the base of a right-angled fpherical triangle be conftant, the increment of the hypothenufe: increment of the bafe: the fine of twice the hypothenufe: the fine of twice the bafe. He next proceeds to the variation of oblique-angled fpherical triangles; and here the reader will find an investigation of all the different cafes. This is a fubject of great confequence in aftronomy, where it is fo frequently required to find the cotemporary variations of the different parts of a triangle. If a fmall variation of the fan's altitude be given, we may hence find the cotemporary variation of the time, or the contrary. The diameter of the fun being alfo given, the time by which his rifing is

acce

accelerated by refraction is known. If a fmall increafe of the fun's right afcenfion be given, the correfponding increase of his longitude will be given. In fhort, in the prefent improved fate of aftronomy, this fubject is of the first importance.

The author concludes by fhowing, how the properties of plane triangles may be deduced from thofe of fpherical, in thofe cafes where the fines or tangents of the fides enter; for, by diminishing the fides of a spherical triangle, fine limite, the triangle approaches to a plane triangle as the limit, and the ultimate ratio of the fines or tangents of the fides will be that of the fides themfelves; for instance, in a fpherical triangle, the fines of the fides have the fame proportion as the fines of the oppofite angles; and when the fides are diminished fine limite, we get the proportion of the fides, the fame as the proportion of the fines of the oppofite angles, which is the property of plane triangles.

In this work, the author has confined the plan to whatever may be useful in its application to fcience; and he appears to have comprehended in it every thing which can be neceffary. for that purpose. Moft Treatifes are either too fhort, or are extended beyond the bounds of what may be fufficient for practice. The work before us, we can recommend, as comprifing all that can be generally useful on the subject, and no

more.

ART. VII. Archæologia, or Miscellaneous Tracts relating to Antiquity. Vol. XIII.

(Concluded from p. 74.)

XII. Copies of Two Manufcripts on the most proper Method of Defence against Invafion. By Mr. Waad. Communicated by the Rev. Samuel Ayjcough, F. A. S. in a Letter to the Rev. John Brand, Secretary. Read March 2, 1797.

THE

HE author of thefe MSS. who fucceeded his father, a Yorkshire gentleman, as clerk of the council, was knighted by King James I. at Greenwich, May 20, 1603, and made Lieutenant of the Tower, having been employed on various embaflies to Spain, Denmark, Germany, France, in '1 586, and Portugal during the interregnum. He has fhown much good fenfe in these papers, which may be confulted with advantage by those whom they more immediately concern. They are

happily,

happily, however, now become less interefting, than at the time when they were read to the Society.

XIII. Copy of a MS. in the British Museum (Harl. MSS. 6844, fol. 49) entitled, " An Expedient or Meanes in want of Money to pay the Sea and Land Forces, or as many of them as fhall be thought Expedient without Money, in this Year of an almoft univerfal Povertie of the English Nation." By Fabian Philipps. Communicated by the Rev. Samuel Ayfcough, F. A. S. Read March 9, 1797•

This MS. bears date July 4, 1667. After mentioning the brafs coinage of Elizabeth, and enumerating the various limilar expedients, which the Spaniards, the Dutch, the Swedes, the Genoefe, Turks, &c. had on different occafions adopted, this writer recommends, as a remedy for the urgent neceflities of 'the times, that "fome fmali moneys be made of brass or tin, which other nations have but little of, and by a late invention will very much refemble filver." The deficiency of cash in modern times is more readily fupplied by bills of exchange, A fhort account of this projector, Fabian Philipps, is fubjoined in a note, extracted from Wood's Fafti Oxon.

XIV. Explanation of a Seal of Netley Abbey, in a Letter from the Rev. John Brand, Secretary. Addreffed to the Prefident. Read Jan. 26, 1797.

་་

The infcription of this feal is "S' BEATE MARIE DE STOW'E SCI EDWARD," or Sigillum beate Marie de Stowe fancti Edwardi." Edwardflow occurs in Tanner's Notitia Monaftica, as the old name of Netley Abbey; and "Stow" fignifying "place," Mr. B. thinks,

"that Edwardflow latinized upon this feal by Stowe Sandi Edwardi was probably the original name of the monaftery, and that this was its first feal, reprefenting the Virgin Mary and child with St, Edward, with uplifted hands, kneeling before her."

"This famous abbey, diftinguished by the feveral titles of Netteley -Lettely Edwardflow, or De loco S. Edwardi juxta Southampton, was founded in the year 1239, by king Henry III. for Ciftercian monks from Beaulieu, and dedicated to St. Mary and St. Edward.”

P. 194.

In the fame plate with this feal are given drawings of two others, much mutilated, of this abbey, under the name of Lettely Abbey, appendant to a deed, dated 3 Edw..III.

XV. Ex