impoffible quantity 1, remarking their ufe in phyfical aftronomy; and to express an Arc in terms of the fines of multiples of that Arc; and here the author fhows, that if z be any Arc, z = fin. z — { - } fin. 2 z + 1⁄2 fin. 32 —, &c. ad

infinitum.

The author next proceeds to Spherical Trigonometry; and here he begins with definitions, and the defcription of fuch circles of the fphere as the fubject neceffarily requires; and thefe, the reader will find, are explained, and their properties inveftigated, 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 usually met with in works of this kind, but which are frequently found ufeful in removing the ambiguity which arifes in the folution 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 much smaller 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 memoy. The author has purfued the fame method for a quadranal triangle; that is. a triangle having one fide = 90°, which he has shown can be refolved by the circular parts; and he ha arranged the equations in a table accordingly. The ambiguous cales 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 rife. The neceflity alfo of attending to the figns of the quantites made ufe of; that is, of the fines, cofines, tangents, &c. 's fhown in feveral inftances. Some further proportions of ight-angled triangles are added; and fome properties of olique-angled triangles are demonftrated, from letting fall a perpendicular from the vertical angle upon the bafe. The foltion of oblique-angled triangles is next given; and here the realer will find all the cafes inveftigated in a clear and fimple manner, and the rules very plainly flated; fome of which invefligations are new and to thefe are fubjoined a great many affections of fpherical triangles, which will be

found

found extremely useful 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 here he has chofen fome examples in aftronomy; among which, he has given a dire& 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 investigation of this is first 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 easy practical rule for a complete folution, has done an important fervice to the navigator, to whom this is principally of use. At land, 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 obfervation 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 conftruction and ufe of all aftronomical inftruments, in their latest state of improvement.

The variation of spherical triangles is the next, and last sibject of this Treatife. Cotes was, we believe, the first penon who wrote any thing on this fubject; it was published in the Harmonia Menfurarum, under the title of De eftimation Errorum in mixta Mathefi. The author has here first confdered the variation of right-angled spherical 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 tafe of a right-angled fpherical triangle be conftant, the incement of the hypothenufe: increment of the bafe :: the fin of twice the hypothenuse: the fine of twice the bafe. He next proceeds to the variation of oblique-angled fpherica triangles; and here the reader will find an inveftigation of all the different cafes. This is a fubject of great confequence in astronomy, where it is fo frequently required to find the cctemporary va riations of the different parts of a triangle. I a fmall variation of the fan's altitude be given, we may h:nce 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 ftate 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 fpherical 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 themselves; for inftance, in a fpherical triangle, the fines of the fides have the same 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 short, 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 Mifcellaneous 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 Ayfcough, 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 1586, and Portugal during the interregnum. He has fhown much good fenfe in these papers, which may be confulted with advantage by thofe whom they more immediately concern. They are

happily,

happily, however, now become less interesting, 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 alm ft univerfal Povertie of the English Nation." By Fabian Philipps. Communicated by the Rev. Samuel dyfcough, 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 fimilar 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 "foms fmall 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 cath 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 Faiti Oxon.

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

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

"that Edwardstow latinized upon this feal by Store San&ti Edwardi was probably the original name of the monaftery, and that this was its fft 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 Netteleg -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

XV. Explanation of a Seal of the Abbey of Lundores, in Scotland. By the Rev. John Brand, Secretary. In a Letter addrefjed to Owen Salusbury Brereton, Efq. Vice President. Read May 11, 1797.

From the mutilated infcription of this feal, which runs thus, Sigillum Sancte Marie et Sci Andree de Lund***,' the VicePrefident had fuppofed it referred to the parishes of St. Mary at Hill, and St. Andrew Hubbard, in London. Mr. B. however, fupplied the defective letters res, part of the R being fill visible, and fhows it to have been

a Scottish feal, and most probably the first and original one of the rich Abbey of Lundores in the foreit, on the river Tay, by the town of Newburgh, in Fifefhire, founded by David, Earl of Huntingdon, brother to William, King of Scotland, on his return from the Holy Land, A. D. 1178, for Tyronenfes."

He fupports this appropriation, by copying from the end of the fecond volume of Dugdale's Monafticon, the charter of foundation of this abbey, and concludes by obferving, that "Lundores was erected into a temporary barony by James VI. A. D. 1600, in favour of Patric Lefly, fon of Andrew, Earl of Rothes.

XVI. Copy of an Original Inftrument dated 25 Nov. 1449,. concerning the Church-Yard of St. Mary Magdalen, in Mik Street, London. Exhibited to the Society of Antiquaries, by Thomas Loggen, Efq. Read March 23, 1797.

This inftrument, which is in Latin, is founded upon the depofition of a Mr. Robert Sheffield, clerk; the most remarkable part of which is, "that there ftude a croffe in and uppon the fame voide grounde of the height of a man or more. And that the fame craffe was worshipped by the parifhens there as croffes be commonly worshipped in church-yardes.

XVII. Copy of an Original Letter from Queen Elizabeth to the Earl of Warwick. Exhibited to the Society of Antiquaries, by Peter Renouard, Efq. F. A. S. in whofe Family this curious Paper has long remained. Read March 16, 1797.

This letter is endorsed in a different hand, 4 July, 1563, and relates to a fupply of "men, money, and vittell," which the Queen engages to fend this nobleman, to enable him to keep poffeffion of Newhaven (Havre de Grace). But it appears from Holinfhed, that though the garrifon did actually receive the promised fuccours in less than a fortnight, they were not

able