I N V 



invocation fo clofely to tlie propofition, tTiat he ftems to- in- 

 voke }iis Mufe for the whole work. But Virgil, on the con- 

 trary, only requclls his Mufe to furnilh liini with a part of 

 his fubJL'Cl ; and even mentions wliat particular pan it i* he 

 dofircs. After propofing his nuittcr in all its extent, lie begs 

 the Mufe to acquaint him witii the caufe of it. Sec VnQ- 



P0.SIT10K. 



As to the deity invoked, the fame author obferve?, that 

 it mufl always be, either the divinity that prefidea over 

 poetry^ in general, or that whicli prefidcs over the parti- 

 cular fubject of the work. Ovid's invocation, in the Me- 

 tamorphofcs, is of the latter kind ; and fo is that of Lu- 

 cretius ; thole of Homer and Virgil are of the former 

 kind ; they only invoke the Mufes ; and thus they ditlin- 

 guifh between the divinities who prefide over poetry, and 

 thole who prefide over the aftions of the p .em, and have 

 parts in it. 



By t!ie way, it may be obferved, that the deities invoked 

 are not looked on, even by the poets themfelves, as divine 

 perfor.agts from whom tiiey expect any real affiftance. 

 Under the name of Mtife they wilh for the genius of poetry, 

 and for all the qualities necefiary for the execution of their 

 defign. Thefe are mere allegories, or manners of expreflliig 

 themfelves poetically ; juft as when they perfonify and make 

 gods of fleep, of reft, fame, and other natural and moral 

 things. And thus the iVIiifes come to be of all ages, coun- 

 tries, and religions ; tUere are Pagan, Chriftian, Greek, La- 



ics, ana religions ; ti*e 

 1, and Enghih Mufcs. 



INVOICE, or I.svoYCE, a lift, or account of commo- 

 dities, with their value, culloms, provillon, charges, &c. 

 fent by a merchant to his faftor, or correfpondent, in another 

 country. 



IsvoiCE, Book of. See Book. 



INVOLUCELLUM, in Botmy. See the foUowing 

 article. 



INVOLUCRUM, a term adopted by Linnius, at the 

 fuggeftion of Artedi, for the row of fmall leaves found 

 at the bafe of the umbels in moft umbelliferous plants. 

 That of the general umbel is termed the General Involu- 

 irum ; that of each partial one a Partial Jiive/ucrum, in 

 I..atiu Liva/uc:llum. The fame word is ufed for what might 

 he more properly called Brafteas in Euphorbia and other 

 j^encra, and Linnxus has probably erred in reckoning the 

 invo'.ucrum a fpecies of caly.^ in any inflance. In ferns, 

 however, we retain this appellation for the membranous 

 covering of their fructification, in preference to a newly in- 

 vented terra indufium. 



INVOLUNTARY Blfckarge of Urlnt. See IscoN- 

 ■iIM;NtE of Urine. 



IsvoLUVTAJiY jlTotlon. See Motion. 



INVOLUTE Cliive, in the Jil-itr Geometry. See 



EVOLUTE. 



INVC-LUTION, in jllgelra, the raifiug any quantity 

 from its root to any height, or power, afligned. See 

 Power. 



Any quantity multiplied by itfclf, the product \& the 

 fquare of that quantity ; the fqnare being again multi- 

 plied by the quantity, produces its cube. See Sulare and 



ClBE. 



Thus ]£ a + b were to be raifed to its fecond power, it 

 will produce a a -r z a b -r b b. 



And if it be involved again, or if that fqiiare be multiplied 

 by the root, the cube or third power will be produced ; •i>;'z. 

 c a a -i-^aab-'riabb + bbb, &c. fo tliat the invo- 

 lution of any compound quantity is performed by a con- 

 smnal multiplication of it by itf- If : if the powers oi a — b 

 aj-e r-quired, th?^- will be found the lame as thofe <ji a + b, 



J O A 



N;cept that ttie terms in which the exponent of b is an o63 

 number will be negative, becaufe an odd number of multii 

 plications of a negative produce a negative ; or, in general, 

 tlic terms of any power of a — b are pofitiw and negative 

 by turns. See Biso^l \L 'lieorfni. 



If a quantity, confiiling of three or more terms, is to be 

 involved, it may be diftirguilhed into two parts, whicli 

 are to be raifed to any power in the fame manner as a bi- 

 nomial : and then by the fame rules, the value of thofe 

 compound parts may be fubftituted in their ilead : thus, 



<r+TT"fl' = a + b + (■]'- = a + I:* + 2 c X a -t b + ,' 

 = a'- -^ 2 a b + b- + 2 a e -^ 2 b c + c'. 

 ArAa -i- b -r e\' = a + b^ + 3 <^ X a + bY + ,? "t" X ~aT b 

 + ;:' = «"■ + 5 ,r J + 3 a *- -(- i' + 3 a' c + 6a be + 

 3^'^ + 3"^^ + 3> b c- + c''. The reverfe of invo- 

 lution is called evolution. See E.\.rK.\criON of Rota-s, rjsd 

 Root. 



INAVARD Flanking Angle, in Fcrujaallon. See AxGi.E. 



.TOACHIM, in Biography, a celebrated Italian monk,, 

 was born at Celioo, near Coienza, in the beginning of tin? 

 I2t}i century. After he had received a common education, 

 hi? fatlier obtained for him a pod at the court of Naples,, in 

 which he ferved fome time. After- this he took the refo- 

 liition of viliting the holy places in PalelHne, left Naples, 

 without communicating his dciign to his father, and reached 

 Conllantinople, where he made a fhort ftay. While in this- 

 city he was fo alarmed at the extraordinary mortality pro- 

 duced by a pcililential diforder, that he made a vow to 

 renounce the world, and, having aflumcd the habit of a her- 

 mit, proceeded barefoot on his journey. On his return h^ 

 entered into the order of Cillercians, and founded feveral 

 monafteiies, which he governed with great difcretion. Ha 

 was regarded by many of his contemporaries as a prophet, 

 and feveral of his predictions were printed, in a book en- 

 titled " The Everlalling Gofpel.'' He was author of other 

 works, particularly of " A Harmony of the Old and New 

 Tellament." He died in the year 12C2, at the age of go. 

 His followers wilhed to perfuade the world that miracles 

 were wrought at his tomb, and applied to the pope to ca- 

 nomz.e him, but this was rejetted on account of the numev 

 rous errors contained in his works. Moreri. See Joaciiit 



iUTES. 



Jo.vciu.M, GsoRGE, a Celebrated mathematician and aftro- 

 nomer in the lOth century, was born, in 1514, at Feldkirk, 

 in the Tyrol. He was educated at Zurich, where he foon 

 difcovered a tlrong inclination fc^r mathematical purfuits, in 

 the elements of which he was initiated at that place. After- 

 wards he went to Wittcmburg, where he profecuted his 

 ftudies with ardour and fuccefs. In the year 1^55 he was 

 admitted to the degree of M. A., and in two years after- 

 wards he was appointed joint profefior of the mathematics 

 with Reinhold. He was a zealous and able defender of the 

 Cbpernican fyllem of the world ; went into Priiffia, where 

 he placed himfelf under the tuition of Copernicus himfeif, 

 and became the editor of his great work, entitled " De 

 RevoUitionibus." While .loachim continued in Pruifia, to 

 render ailronomical calculations more ready and accurate, 

 he began an ekborate canon of fines, tangent.s, and fccants, 

 to llfteen places of figures, and to ever)- ten feconds of the 

 quadrant. He did not live to accoaiplifh this great work, 

 but the canon of fines for every ten feconds, and for every 

 fingle fecond in the firft and lad degree of the quadrant, 

 computed by him, was publifhcd at Frankfort in 1613, by 

 Pitifcus; the larger work was perfcfted and publiftied' alter 

 his death in J5<>6, by his difcipie Valentine Otho. Joachim 



retuiacvl 



