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p. R 8. LOXD. AM) EDIX. AND M. R. I. A.

•r T«B BOTAL »c*aatn or tagstcm Ar rAmm, a\» or mu aortL AcjiDmr or KUDicn or rasMU i ManBa or t^ mvu. umtimm tctatmt or mubkb; or t«s mtal mcwtt or M;nD«aB or DwaiAmK ; or tat Moru. MnvTT or ouiiuum, *m or ram mmtal M-.»ourr or ■-imwuB or Moon* ; wxMMukt mmciatb or tub ■•TU. «ckBanr or MfmcM ev ltcm) mmcutb or t«i McvrT or avn. ■iiiiiirnw; mmaa or tmb ■ocutv or tiib an- TKTtBiw or tcvfuiwi oT Tvl aWMMeu. ■onarr or umwor, or tmb mibbiiuwicm. Mcnrr or lomdox ; or mB utmKJOi umatABUB Moarvi ■owoBiBr mbmbbb or rm unaABT md rwu«onnc4L MM-narT or mew vobb; or tub MVPOaacAL fooBTT or warn Taast bv vmb utvubt *av mmmmmtcu. MrntrT or vTBaorr ; or tbb nnuMomicAL


T«B BovAL tamDii. BB» rmvmcAL matnm or ■WBBiiwia ; or tas ACAoaar or natcbai tanvu or randHitiraiA ; or I or BATvmAL BimNiT or BBatnt or taa »ATt-BAL airroBT toaBrr or rBAMKMBT; or tiik ' I— Bi or TWB BOTAi. nantiBiicAi. wcatrr orooBswtix, ako or ras nauMoraiCAL

TM* AanvAiKS or












IICHESTER. or Uticmnrtm. Um UMU of Pu>. Jy»i«;bowiyh and OMrfcct lawn of Engluxl, in QwiMMlriiiit. It la atmad on the river Yto, ottt wWdi Ikn* H a Kofw teUg* of two knw wdiM, md en*to«f ftw «n*u, whicb mbvt tnflfcuatlr twih.

n* a^ wwai g tkmh mnaidx k dt^aittd to

■tMiy.MwdhaaaB nrmnnttcw«rgOtb«ttiigfc, cww •^""yy "J— ' «««— - Thcraarealaoplaotaofwerw aUpftrlkatfaaaMark The aaaiaaa wm frnMriv held kanandarfepMaalgmtadby Ed*. •-: " ,f^ afvMv^haUlMninrBlAaw.v go'

and %i^|l««aMr. Tha OMntjr eoort-hou*.; u gooi """Jf; —■?>»«*»■«» eowity yaot. whiri> baih aMMrHowM<rapla& Oypaaita to k. oa the other ddt ?ill^ "TJ*^ »>» fWMta of tha ancient hoapital

"^ ^^^"H?*** *"« *•»•• »V Waikn, uSida or Dacna. Ut tha aMarmawM <^ pOgrina or noor uaidlaa. Thar* b abe to be aaaa aera tha leniaiM

paMwa MadMf to tha WMto>hall nonoenr.

Whan tUa part aftha oammry wu in the im |j ,

of tha Wiwwi. Ifchwtar waa one of thdr yriSpSlmZ l<«M. awl waa hrMtd with a aiiia« «d aSdaan

Sd- SsrSLTl?"'*'?^::*^ '*-—

birtit haa bean awnadcd by a hmB Thia town ia ftoMw aa bafai tha y 9t the jaiilnalaJ Refer Bacon Tht ftl- jwHn, k tha |.af iKlw of thelnrgh and paHth Or

N««b««riBknbk«U boui Naniher af faaffiee Do. ein|<U>]rod in tnde PofMlatioa , . .




*1IT I.

"«*»» ?r S^mrrmUUn ; and tha maWmUt. »ol. «tii. p. 5 If?, la a a(»>poct and markct^to^n of

I L F England, in naOjti Deroo. It U dtuated on the llft««»b.

thaBftaol (Aaaari. The town oonuine a number of goea bown for the acoonunodation of bathan, which attend alonf the tide of the harboor, and for nearly a "^ *«>«fcfww«i#il. whan there i. . good pebbly and caovniam bathfaig macfainca. The church intbeinerpartaftbetown. It large pUin

, ■•" eontain* a hamUone nonuncnt, ercct-

Ki, a( the national espenoc. to the meoMiy of Captain Bowen.

The haibonr of llftacomhe rrtemblca a natonU bn. Ml. Mirrawidcd by tnuo ba^fcU. cmmd with fo- IT!!: J^ ~^ aacaatfTn a aemidm.Ur .weep on three aide*, and on tha north Mde a maM of rock pro. Jeeta nearly hJf way acraaa the mouth of the te^aa.

mdprnlecta the little cove fWm the iMrthcrn tempeata. TUa rodE run almoat to a point, yibr^ a IL'ht bouae. tnonbltog a churrh ; i«l. » Aioog the aide of

!5! ^ ""i^" *' "^' '*" '•'• '*'''* '^'w**

latfrentnLommiu .. ... -.i^ opaning of tlie harboor nma an aniOoal pi«r, jwlidauJV cMutructcd, to nral rmttbeaoonnalationoftheMnd; ao that by the joint iiililiiitiof the natural barrier, and thu piac* oT ma. S'iLT? * !*».»«»• burthen may ride comdetelr land-locked, and of ooone perfectly aafe fton Vo the vMltnoeaf the weather. Before the year 17SI. thia Bjar WW «30 feet long ; but having bean deatiwed by the wfnd, an act of pvlianient wat paucd for repair' tag and enlarging it and the harbeur. It waa partly rebnUi. lengthened, and enlarged, in 1700, by Sir Bour. chief Wrty, Bart. '

There a daily intercourw, by mcana of a packet which auU every other lid*, with the town of Swan^ tea, on the oppotito coa^' '■".■.».

One ^jchA near i, , „,„,„„ bouae,

wectod^ Sir Bourch miuanil. a

taq. ia taotifuny »ituati J un aa Huiuenc* about three mtlet to the caat ot' the town.



ll^ri^M The reoelt which belong to Ilfracombe ore chiefly

I employed in the coasting trade, in conveying ore, corn,

IDiBob. jtc. from Cornwall and Devonshire to Bristol, and also

^*'Y^"^ in fuhing ,

The following wai the jKjpulation of the town ana

pariah in 1811. ,

Number of houses *34

Number of families 434

Do. employed in trade I '8

Do. in apiculture 57

^ PopuUtion 1934

See Polwhele's Survey of Devonthire ; Mason's Ob- tervotiont on the IVeitern tounlies ; and the BeaiUiesof EmgUnd and Wales, vol. iv. p 267. ILFAC Passion. See Medicine. ILIAD. See Homer, vol. «. p. 97. ILLE AND ViLiJiiNE, is the name of one of the north- west departments of France, which derives its name from the Ille and Villaine, two rivers which unite at Rennes, the capital of the deprtment. The soil of this ileparU ment is in general ill-fitted for culture, nevertheless be- low Rennes and St. Malo, corn, hemp, and fruits, are produced in abundance. At a short distance from Ren- nes, is the farm of Prevalais, where the butter is made that is so famous in every part of France. The in- land commerce of this department consists principally of its natural productions, which are corn, lint, hemp, wood, cattle, butter, mines of lead, oysters, and fish. The department contains 7185 square kilometers, or 3&\ square leagues. The forests, of which three-fourths belong to the nation, occupy 54.000 or 55,000 acres. . The contributions in the year 1802 were 421,093 francs; and the number of inhabitants is 488,605. The prin- cipal towns are.


Rennes 25.904

St Malo 9>147

Vitr^ 8,809

Fougeres 7,297

Redon 3,783

Montfort 1,118

ILLEGITIMACY. See Law. ILLINOIS Territory is one of the northern of the United States of North America. It derives its name from the river Illinois, an Indian word, which signifies a nan of full age, or in the vigour of his years. gj^^ This territory lies between 37° and 49° 37' of North

Lat. and between Long. 85" 45' and 95" 6' West. It is alwut 870 miles long from the Ohio to the northern line, and has the following breadths, 650 miles, 200, 150, and 50. It contains 200,000 square miles, exclu- nve of part of the waters of the lakes Superior and Mi- di igan. Bounduio. ft is bounded by Upper Canada on the north, by the Indian territory on the east, by the river Illinois on the south-east, and by Louisiana on the west, from which it is separated by the Mississippi. Dinnoo "^^^ P*""^ °* ''''* territory which is settled by the

■od popoU. white people, is divided into two counties, St Clair and (wn. Randolph ; the first of which contains nine towns and

5007 inhabitants, and the second three towns and 7275 inhabitants. In 1800, tlie whole population was 215, and in 1810 it amounted to 12,282. Town*. The principal towns of the Illinois territoiy are Ka-

laskia, the capital, which is situated on the river of the

I I- L

same name, and contains 100 well built houses, and niinolj, 0'22 inhabitants ; Cahok'a, situated on the southern side lUuminiui. of Cahokia Creek, and containing 711 Inhabitants; and ^T^f""^ Goshen, the capital of St Clair county, which has 1725 inhabitants. . . .

The chief rivers are the Mississippi, the Illinois, the Wabash, the Ouisconsin, the Fox river, the Chippeway, the St Croix, the St Louis, the Winnipec, the Do\e, the Seseme Quian, the Kasaskia, the Ausvase, and the Little Wabash.

The principal lakes are Lakes Michigan, Superior, Rainy Lake, Red Lake, about 60 miles in circumfer- ence, and nearly round ; Lake Pepin, about 20 miles long and 6 broad ; Lake Winnebago, 1 5" miles long and 6 wide ; and lake Illinois, about 20 miles long and 3 wide.

The country between the rivers Kasaskia and Illi- nois, which is a distance of about 84 miles, is a rich and level tract of land, terminatbg in a high ridge. The last of these rivers is bordered by fine meadows, and the soil of the country is in general of a very su- perior quality.

Fort Massac, which was built by the French on the west bank of the Ohio, is a port of entry ; and in the 4th quarter of 1 803, foreign articles were exported to . the value of 17,320 dollars.

The principal mines of this territory are those of Natural copper and lead. On Mine river, a western branch of history, the Illinois, there is a rich copper mine. On the south shore of Lake Superior, there are many mines of pure copper. About 9 miles from the mouth of Iron river there is another copper mine ; and the same metal is also found in great quantities on Middle Island, nine- teen leagues north-west of Iron river. The purest lead ore is found in immense quantities on the banks of the Ouisconsin.

The banks of the rivers abound with buffaloes,- deers, elks, turkeys, ducks, teal, geese, swans, cranes, pelicans, pheasants, partridges. The sturgeon and the picannau, and plenty offish, are found in the lakes and rivers.

The vegetables of this district are tlie oak, hiccory, cedar, mulbi-rry, hops, dyeing drugs, medical plants of various kinds, and excellent wild grapes, from which, in the year 1797, the French settlers made 110 hogs- heads of strong wine.

Before the year 1 756, the French had settlements at History. Kaskaskia, Cahokia, &c. but they were at that time driven out by the British, who held the country till the revolution. In the year 1780, there were 12 Indian tribes in this territory, which were estimated to con- tain 8300 fighting men. See Morse's System of Geogru' phy, Boston, 1814.

ILLUMINATI, a secret association which existed in . Germany, some time previous to the French revolution, and which has been supposed to haVe been connected with the masonic institutions on the continent. ' This association was founded in 1 775, by Dr. Adam Weis- haupt, professor of canon law in the university of In- golstadt The professed objects of the institution were to introduce more enlightened ideas of government, to disseminate a knowledge of the sciences, and to pro- mote the interests of virtue ; but its members have been accused, not without some appearance of reason, of in- culcating speculative opinions, equally hostile to the principles of sound religion and social order.

Soon after the commencement of the French revolu- tion, the attention of the public was eagerly directed towards th^ plau and objects of this association, which


Umb iDdttd IM looffr exutcd, but whoee merabcn ware Mimmwd to hsTe extirottd no iooooaidcrablc par- tioa of activity uxl influenor. in pndmaag the pccu- •■m1 dircetinc the political erenta of the

mt l—yai, aiMl dircetm* the political ervnta of the tiaaa: aad tb« work* of the AhW Bamirl and ProTrv ■r Babinn, in which the Mcrct view* and active rxer- liaat at the orrier were auwght to be developed, were yamaad with t'straordinarjr latum and aviditv. It ■eaaa now, however, to be prcdy gcncralljr acsiiow- ledgcd. that thaae, ami other author*, were indoeed to Mchbe to this iortilaliea an eitent and an influence, vhich in reality it nevar BOMiaHd i and that, in pwti> cular, the cecret nachiOBiam and wicked pr«cticca of llle ill— lyKti were excaaaraljr aMfni6ed hj tha heated ' of the Franch pncat. and the hoocat cradu-

Wmj af Mr iHaDQ|MHHn ONnvfynan. ine maaory ot 1l» artltr of the JOwMili iadMd is involved n much


ehaearity ; aad thia ihwinalaiice. which haa led Mhape natMBOjeae^gfc^loeverrMc iu inffuencc. (obm* tMkiB arv wmpiiflia) aArda to oar ndnd no mean evi>

dweeoniaiBaigBilcaaoe; Time,h0wevar, ha< ^

nil In rtiaiinMi rtia hMgiuM^ iin|iiwHnre nf i MMeandeliaa; aodtfieMlowinf ahaetatalenimt cun* laiM llM fOMnl Ma«k of all the mlfamaliaa we have been able to obtain, reiaiivetoihiaoMelMMertinc.bat

■I oaBoaile BBe rf p^hv •*• pnTanod hv Ike ralaea, AfMd. k wa»Mw.hnihe light of iriiewrfwM 4iMl Ihve MMitat piilaitiini. omb which they can. «*«^ Ite the HMte «r«JHir cMI «Ml nlMoM a«^

\ I L M

surd and barbarous policy. Men of enlightened minds ininainiic, could not fail to look with abhorrence upon regulations, Umimttr.^ which were calculated to check the natural progress of ~ """^ knowledge, and would readily endeavour to concert the means of evading; the existing laws. These means, however, could only be concerted in secret ; and to this simple origin, we believe, the institutiun founded by Weinhaupt may truly be ascribed, however widely tKe conduct of Its members may have alierwards deviated fWim the anginal object. There is undoobtedly some* thiqg dangerou* in the very nature and oonatituiion of such a secret association, however pure the intentions of it« founder* may have been. Secrecy implies Mtne- thing illegal in the object ; and >uch a society being un- der no regular controul, its views may easily bettime •nlargcd, and its ii.fluence perverted to improper purpiaca. Some sealoua enthusiasts among tlie il- Inminati, may have contemplated the possibility of diractiM the eiisting govrmmirnts, by means ol' the puw«Au iMt pacific influence of a secret ussociatiiin ; other*, imbued with the a)>*urd theory of the iiifirute perfectibility of the human mind, may have cnn.-iilered ' such an ins^totion as calculated to promute tlieir ro> maotic ami MMttaitvuble views ; while a third class, perhaps, cawaiating of men more cunning than enthu. aiaatic; Ml of ambition, hot destitute of principle, may hare leoked upon a aociety ao constituted, as best adapt- ed to the eeoeoction of their wicked designs. Yet the aoeiely ef tlieflliiiMaali nercr seems to have acquired HJ ailMMisa iafMBee: nor doea it appear to luve poaaaaad aay nnlfloiitioas bcrend the limits of Cier- ■MBT. Tbraogimrt tiw whole of that large empire it fmwwd DO eitiauwliuwy or pennanent eflecu , and lbwy«n alter the aa^pteaaioa of the order in 1787> it WM nearly fttgottaa in the vtty country where it hod boHlad ao epfeMNnl emialeaea It was chiefly on ac« eoaot ef ita sopoaad ininaiiei in producing the catai tropbe of the Frfnch Revoiation. that the ashes of thia ahart-lived asancfatiaa were raised mftom the charnal* howe «t obllvioa, and a dcgrar or poathumous oal*> brity cttdkmi upon ita prooeeilfnga Hie aappoa- , Iwwevar, ia entirely anaopportad by eridcner, haa been sidBdrntly refuted by M. Mounier and

iaoUtc thimselve* mm bom. la eaerciae a •peciaa of lollecw of their sabUta, by ■gain t the til laiinrroP antidoia arigBotanoe la the Bat it waaaaeaay aHttar lo Mmko. ar to keep the aiek edi, in eoaatrv whose ia Hn*) iirigm, and spoke one sFv^ the BMDs of


■i-n they

■I- a .f( Uy an<l

biguiaJ •re; and 'eiad

scattered ra^ ef the

'Tiiant the II "

those npoa whooi 1 ay wofeptaeaiL md

fell the toticiiaia lanoaeoftha

oTHleratarr than ia Bav*. of the elector , It io pncisfly in ' waahof such ab- I

But wbatfTtr iMy have been the eitent, influeaoe, and roal otfjecta of thia societv, there can he no doobt Aat Ita Bonitilatleii waa Ukgal, and the opinions and practiMa of ita lawbin higftly dangerous to civil and lallgioa* guiaiMl The aappreaaiun of the order.

thoraAire, waa Jisslifcbis opoa every prindpir of right Md wpadlriHTP. A great maas of pubbcation* has sp- peered apon the Buhject of this article ; but the suD« MMC* ofthe whole frill be rannd in the works of the AbM Bamel, Profeaaor Robiaon, and M. Mounier. (;) ILMINSTER, a market town England, in the eoanty of Samtntt. It has a low but heaitliy situation.

at the iaiei aactiwi of two great road*, oi e from I.oiidon to Taunton, and the other fVnm Bri>tol to Exeter. Many of the houses, which are principally arranged along theac roads for the length of a n.iK-, are good stone and brick bu.lding* ; but the Krcatir nunil>er are built from old walls, and are covered with thatch. The church, which is a fine CicAhiC buildini;, is I'M Icet long and SO wide. A handsome quadrangular tower riaes in the centre, sunniNiiitcd by I '2 pmnactcs. tii the north tr;. . ' , ,,( urnxb,

erected to t: and his

wife, who fouudcU Wadham College, UxiorJ. In the


!•■■• centre of the town is a new built market-house and I th;inil)1es. An excellent free school whs foundeil here

iwi!d2 »" ''''"• ''y Humphry WaI<lron and Henry Greenfield, T|** !_"- nnJ well endowed. The cloth manufacture once flou- ri«he«l here in n very great degree, but, though still considerable, it ha« greatly decreased. Ilminster is iturrounded with fine orchards, and very extensive pros- pert* arc commanded by theneijfhlwuring eminences.

The following is the abstract of the population for the town and parish fur 1 8 1 1 :

^ Number of houses S65

' Number of families 403

Families employed in trade 231

Do. in agriculture 121

Males 1022

Female* ' . - 1138

Total population 2160

See the Benulies of England and Wales, vol. xiii. page 533.

IMAGE. See OpTicsi

IMAGINARY Expressions or Quantities, or im- possible quantities in Algebra, are such as have the sym- bol t/ I in their analytic expression. They are so called, because the square root of a negative quantity can have no real existence ; for whether a quantity be positive or negative, its square is a positive quantity. , The origin and nature of imaginary quantities have

been explained in our article Aloebha, § 189 193. Tliey are there shewn to be of two distinct kinds, one which is altogether impossible, and can denote no real quantity ; and another, which denotes real quantities.

The first class, when reduced to their most simple expression, have this form a -[- »/ b^, or a-\-bt/ 1, where a may in some cases be = o. These occur, when a problem is to be resolved which from its nature requires that the data be contained within certain li- mits in respect of magnitude, while at the same time, in the particular case proposed, they pass those limits.

For example, if it be required to construct a right angled triangle, the hypothenuse of which shall be equal to a given line a, and one of the sides equal to another given line b; from the nature of the case, « must l)e greater than b ; and if, in the particular state of the diita, a be less than b, the thing required cannot be done.

The unknown side of the triangle expressed in sym- bols, algebraically is j^ (a' i') ; new, if 6 be greater than a, the quanti'.y a' 6' is negative, and the ex- pression for the side of the triangle has the form i/ n'=:nv' 1, which is imaginary. The impos- sibility of giving a significant numerical value to this symbol, corresponds, in this instance, to the impossibi- lity of placing between a given point, and a straight line given by position, a line of a given length, that is shorter than the perpendicular from the point on the line, or, which is the same, of determining the intersec- tion of a straight line, and a circle which lies wholly on one side of the line.

In Geometbical Problems, passing the first order, the unknown qu.intitie3 are determined either by the intersection of a straight line with a curve, or else by the intersection of two curves : Now, although it may be possible that the conditions to be fulfilled in a pro- blem may be all satisfied at once, yet in many cases there will be limitations of the data ; for example, by one condition a straight line may be required to be of a given length ; and by another, that its extremities


be on the circumference of a given circle. These can only be satisfied at once, wh«n the straight lin^ is less than the diameter. In like manner, one condition re- quiring that a straight line touch a circle, and another, that it pass through a certain point, can both be satis- fied only when the point is without the circle. When the data of a problem are in this way limited, as often as they cannot be all satisfied at once, the incongruity is indicated geometrically by their being no intersec- tion of the lines, which should meet and determine the unknown quantities ; and algebraically, by the impos- sible symbol ^ 1 entering into their values, and in such a way as not to admit of its being eliminated. The presence of the symbol >y 1, in the aJgebraic expression for a quantity, serves not only to shew the impossibility of finding that quantity in the particular state of the data, but it also indicates the boundary which separates the possible from the impossible cases, and thus determines the greatest and least values that can be given to the different quantities concerned in the problem.

For example, let it be required to find a fraction which, together with its reciprocal, shall be equal to a given number, and also the limits within which the problem is possible.

Calling the fraction x, and the given number 2 a, the condition to be satisfied will be expressed by this equation :

a; + = 2a,

which produces the quadratic equation x* Zaxz= 1, and, this resolved, gives

x = azi=^{a^—\).

From this expression, it appears that the problem is impossible if a be a fraction, positive or negative, be- tween the limits of + 1 and 1, because then a' will be less than 1, and a^ la negative quantity, and ,i/{(fi 1) an impossible quantity. However, if a be a positive quantity not less than -j-1, or a negative quan- tity not greater than 1, (here we reckon ' 2 to be less than 1, and 3 less than 2, and so on), the problem will always be possible, and, excepting the ca- ses fl=:-}- 1, and rt= I, X will have two values, which will be reciprocals of each other, because their product is unity.

It also appears that the least positive value of the

expression x ■{ is -J-S, and its greatest negative value

2, reckoning, as before, that negative quantity to be

greatest, which, independently of the sign, is expressed by the smallest number.

Hence, we learn that no real value of x can be found

that shall make the expression ^ (^H j equal to a

proper fraction, either positive or negative, but that, this expression may represent any positive or negative quantity whatever that is not between the limits of -f 1 and 1,

From this example, it appears that the theory of im- possible quantities may sometimes be applied with great advantage to a very elegant and interesting class of problems, namely, such as require the determination of the greatest and least values of a variable quantity. The general method of proceeding is to suppose, that the quantity to be a maxvuum or minimum, is equal to 8 given quantity; and then to inquire what is the

Imaginaty QaaiUiiitfS.


-,-fltrAt or iMit raluM which thu qiuuiUtr can have,

'raducinffUM imaginary symbol v^ 1 into

i„' formuor.

Tbe seoKul cUm of imaginary exprtnioni, or those

which indicate ml quantitie*. involve the symbol

^ 1 in such a manner, that, by suitable transrona*-

tioRS, it may at last be made to disappear. The two

fbOowhig expratimis are of thia kind, viz.

their aid, laould have been altogether untractable. We Imsgiwy have given examples of their application to the theory /_"""'^ of angular sections, and the investigation of that ele- " ' l^ant property of the circle called the Cotesian theorem, in the Arithmetic or Sines, j 19 i 25. If in the funnuU

(Co*, x4.sin.xv/— I)" = COS. n x-|-sin. m zv^— '1 we write for m, it becomes

~-'l J (Cos. x+sm.*v' 1 j«S5Cos.— + sin.— v^ 1.

Bt takinc the square, and then Main the square root ei each, uie fiiMi m tnudatmeata

and the latter to

wUck an both rs^l qasBlitMa.

Tbe Hciicral cxpmaoa (or the roots oTa cubic equa- tion has the form

*•(•+»•— l)+V{a-^v/— I).

when ha raols ara ail real ; but. nnlike tbe two former, it eaaiMt. bjr am nana, ba tnmSantmd faMo a real al. gafaraic eaprtsiiim, oonsMttng af a loita nomber of ttraM: but ita value nuy ba wand by an infoita sa- rias.oratibleafsiiMs. (Aiasaa.*. f tt?— 4 S9u)

It has bacn movad, in the Abithmetic or Sinks, { 19, that, as bebif any whola nanibii or fraction,

(««t*+ MBuX •—!)-= OBB.»* 4. StO. MXv^— I.

This figrnwla was first given by Da Moivr*. IPUL TrmMM. 1707. and .V/m«/ Ammlfirm, Wb,%) ' caBs it '< a fonaola as rswarkabla Csr its

aodclrr ito gtocva% and CMihty

qurar. »/. drj Famctkma, p. 1 16); awl Laplaaa

Misiilm lu larcatioa as of aqital ia^MVtaMa with tha ■"— "' thcoraaa. (Lr^aas dm Eeakt yanaafei.) As tha «gB of tba square root of a niiMitil| may baaithar .for —I. we nay pat ia (be Cwaaala «^— I UHtaad of ■fv^— 1 : it than bccooMa

(•^'— ■«>.'•— I >"=oaa.8sx— fin. ■i*v'— 1- From this, and tbe farmer npraasiaii. we And. ad.

ditioo and •ohtiactica. that tha i

(•^*+ •«»•»•— •r+(««-«—tfat.*v'—l)* is cqoinJtM to i coa. as x. or raal quantity ; ake, that imaginary wiprwsiwi

' J

(<»-»+•«««.'•—»>■— (cost*— sin. «v'—l)" }


isaqpivalsMttoSsia asx, annthw rml niisntll 1 ■liialiaii akhougb the feiMsntativaa of f«^ *"«»M«"w«. TO- « ooa. >■ X, and JJ sin as x. derr.! I)y themaalvea. am utterly without any trical stgniScaticn. ft is iafMasibli ta analytic rormnbe into the '— jntsga of strict

*/— I

tha symbok ain.xv^-.| and to nothinit that adauta «f gaematri^ definition.

•ion* bavalcdlatbe 'liie^wy'^ mam ITtbTS^

Suppose nown to be indefinitely great, then, xbeingsup.

poaed a finite arc, the arc will be indefinitely small ;

in this case its cosine will be equal to the raxlius, and its sine equal to the arc itaelf, hence, n being indefi« nitely great,

^Cos.x4.sin.x^-.lV=: 1 + y/_l

or putting 00a. a -|- sin. x ^—1 ^ *.


•= IH v^— 1

a and haoea a/eV— .l^ s x^— |.

Now it has been daoBomtratad in the intndnetion to tha artida Flvxioms^ f IS, that % being suppooed in.

initely great, n\v "— •)= N*p. log. », tbarafera

Nap. log. a = xv^->l aod hnca^ c danotiiy tha bmia of the system. (Al.

•mu. |SS5Midf 358) wa hares = e'^~'' that if,

Coa. M 4. sm. X v^— 1 = t

Ls^ im^oarv formula of great value, bOb it atWhite uwlor a Inim form a relation between

Ma eo.ama, sine, «cc It was first by nnlar, and is justly reganicd as one of the nnportant analytic invention* of the last century. Other ittvr.tigations of iMo tbrmula have been given in AaiTHMiTic of :Nnoa, f fp, and Fluxions, } \t\. Ub. serving, aa bafaw. that tha aqnaiv root of a quantity may ba OMHidaiwd as negatiw as well aa positive, wa have (ram the formula,

Coa. X sin. * v'— = «'"'*'"'' '

and from tha two cxpreasions, by addition and sub. traction.


Sin. X

Tbcae fbrmulv, in their present state, arc illusive 1 for the arc and sine, or eo-sine, cannot, by means of them, be founfl, tbe one (Venn the other. However, by ex. panding the exponentials into series, we have, (Ai.OB< aaa, { Si? )

'*'"'* x*»/— I




= 1— ,v^_i—


for Sin. X Cos. y + Cos. 5 Sin. x", (AftiTHMETic of ^T^^^^^ Sines,) the product is equivalent to, w>^-l^





ThcM being «ub«tituted in the formula, we get

*^* '~ 1.2 ■*■

Sin.jr = x --|^ + 1,2.3.45

-, &a

, 5ic.

cxprawions which are altogether free from the imagina^ ry aymbol.

An example of the application of the same formulas to the determination of tnc arc in terms of the tnngent, is given in the AuiTiraeTic qk Sines, § 30. They also admit of various applications to the investigation of rules in plane and spherical trigonometry. (See Le- gendre, Elemens de Geometrie, in App. to Trig.)

Because Cos. * + Sin. x v'— 1 = e * ^"~ ' there- fore, Log. (Cos. X + Sin. x v^ \) =1 x ^ I. Let a- denote half the circumference of a circle, of which the radius = 1, and suppose x = ^ t, then Cos. x = 0, and Sin. * = 1 j hence we get Log. v' 1 = i t 1/ 1, and

4 log. \/~~l

2«- =


This remarkable expression for the circumference of a circle was first found by John Bernoulli. It is of no use as rectification of the circle ; but it shews, that if the expression should occur in any investigation, we may substitute, instead of it, the real quantity 2 r.

The formula, Cos. x + Sin. x y'— 1 = e *v ""*» be- comes, in the case of x = ^ w, yf I = e ' "^ ^"^ ' Now, let both sides of this equation be raised to a power expressed by v^— 1, then, observing that ^ ;r \/— 1 X -^—1 = ^ », we have,


= 0.207879.

This very singular imaginary expression was, we be- lieve, first noticed by Euler.

D'Alembert first demonstrated a remarkable proper- ty of imaginary expressi'ns, namely, that however complicated they may be, they may always be reduced to the form A + B 1/ 1 ; so that every function of an impossible quantity n + 6 >/— 1 has the form A -}- B v/ 1, where, however, B may be equal 0.

I-et r = ^ (u* -f. 6«), and suppose x to be such an

arc, that Cos. x = , then Sin. x ss , where r, Cos. x, r r

and Sin. z will be all real quantities ; by substitution, the imaginary formula a ■{■ b ^ I will be transformed into T (Cos. a + Sin. a ^ I ). If, now, o' + 6' .y/ 1 be another imaginary expression, by a like substitution

it may be trantbrnjed into r' (Cos. x' + Sin. x' ^/ 1),

where, as beibie, /, Cos. x' and Sin x' are all real. Sup- pose, now, an imaginary expression to be the product of the formula; « + A ^—\, and a! Jf U ■•— 1, this, by substitution, will c>e

TT" (Cos. X + Sin.x v'— 1) (Cos-x* .}- Sin.x' ./— 1). By actual multiplication and substitution of Cos. (x + x*) for Cos. X C06. «' Sin. * Sin. x' ; also Sin. (x -j. x')

rr' jcos. (x -f- x') -f- Sin. (x -f- x') >/— 1 j

which has manifestly the form A -|- By' 1- F™m this it is evident, that the continual product of any number of imaginary quantities, a + h ,/ !, 0! -|- y ^ 1, &c. will be an imaginary quantity of the

same form.

_ a -X- b >J I ,-,,. , Consider now the fraction -r—a—, t-- Ihts, by

substituting the trigonometrical formulae, will be chang-. ed to

r (Cos. X + Sin, x ,/ 1 ) r' (Cos..r' + Sin. x'^/— I)

Multiply the numerator and denominator by r' (Cos. x'

Sin. x' is/ 1 ), and substitute Cos. (x x') for Cos. x

Cos. x' 4. Sin. X Sin. x', and Sin (x x') for Sin. « Cos. x' Cos. X Sin. *', it then becomes,

il j Cos. (x— x') + Sin. (x x*) V'— 1 1

a quantity which has the form A + B y' 1 as before. The function (a + 6 n/ 1)" becomes, in like man- ner,

(Cos. X + Sin. x y'— 1 ) « = r" (Cos. « X + Sin. n x \/— 1 )

which is still a quantity of the same form.

By employing the same substitution it is easy t* prove, that the very general imaginary function

(a + i^-l)"' + ''v'-l

is still of the form A -}- B y' 1.

In applying imaginary expressions to analytical in* vestigations, it becomes an important question, what is the nature of the evidence it affords for the truth of the results ? It must be confessed that this part of their theory is involved in some degree of obscurity, lohn Bernoulli and Maclaurin allege, that when imaginary expressions are put to denote real quantities, the ima« ginary characters involved in the different terms of such expressions do then compensate or destroy each other. To this it has been objected, that an imaginary charac* ter being no more than a mark of impossibility, such a compensation is altogether unintelligible ; for to sup- pose that one impossibility can remove or destroy ano« ther, would be to bring impossibility under the predica- ment of quantity, and to make it the subject of arith- metical computation.

Professor Playfair of the university of Edinburgh, has advanced a different theory in the London PUiloi6phU cal Transactions for 1778. He there observes, that im- possible expression.* occur only when circular arcs or hy« perbolic areas are the subject of investigation, and that corresponding to every imaginary formula, there is a real formula perfectly analogous ; one of these is the analy- tic expression of some property of the circle, and the other the expression of a like property in the hyperbola. Thus, supposing that in either curve the semiaxis =1, let c denote any abcissa, reckoned from the centre, .v the ordinate, and x the double of the area contained by the semiaxis, a .semidiamtter drawn to the top ol the ordi- nate, anii the intercepted hyperbolic arc; also let e de- note the number of which Nap. log. =: 1 : In the circle

IMAUlMAKlt i^UArviiiiiao.

QOAT*. ''if*-


= + ' ''

and in the hjrperbolk (aa will readily appear from Fli'Xion*. I 150. Ea 5.)

I art parfiwtly analofpMit in tbcir (bim, ''in tfaefnlMt y^+I be put instead of the ^ ayoibal ^ 1, it will be m—Hiately trans. into tkc wceod. By the fint aetof limnaU. the ) thaavy df the Arithmetic of Siaaa bh^ be invc** d, 4Pd bj the aacMkl. a rwiiifiiMidiiig theory re- ^i^i^ la th» ui uidi—rw of an bypcibaia, bbbt m the luiiieiiiwifiBg iiiiwir*'! ff T*-? T— it rwrn tHI fr irftm tical. and mmbc wuI diftr only in the dgaaaf the tenMi In t>ath« the foaha wtU be alike free froo* the iaa^ gtnary siKH, althganh in the ooe caw the steps by whidi they hare been fbeaid am vaiatelli^hle, and in lb* other, they art perfectly s^aiift. Thii agree- ■eat of two methods so very diftnM ia the discovery of trvth, theiB|[«nioM writer attrihalce the analogy that takes place between the snbfecta of investigation, vhidk is JO daatb iBat eeerr pxysiu of the one may. ieiians, be tfiiilsiied to the other, it hapnai. thai the npirariiai pttfetm ad with dntiiait of amaaiag ikwrna Vm. «• Tft notaa of nfcrcnee othctvwWcb art Mginieal t They peiM oal faMbiwllv a mwhed of deaMatmoag a civiata ptofwfty of tha byptitwl^ and tiltB Itavt as to eanchMle from anekigT ttut the saaw \ also to the circle. All that we are a^


ratio ofmagnUude to be the numerical ratio between the Imigioary ma/niitudes of the two lines and the ratio of potUion, Q"*"'*''**; the inclination of the one line to the other, or the angle ~ » ~ they contain. Again he lays it clown, that four straight lines are in prof/orlion of wtagnttiide and poiition, when between the two last ttiere m the^ame ratio of niagni> tilde and position as between the two firtt. That is, mppoeing a, 6, c, d, to be the masnitudes, we must

hare -j- = —r, also the angle contained by a and b equal

to the angle contained by c and d.

When the conaequent uf the first ratio is the antece- dent of the second, the proportion of magnitude and position is said to be comttnued, and the middle term is a mean proportional of magnitude and position between the other twow From thu it follows, that the middle lann bisects tha angle made by the two estremec.

Theat observations being premised, he gives the fol- bwing theorem as the toun<lation of his theory. Jma- /ftaorv qmmntiiieM oj the rormz±:a^ 1 rrprrtent in Ike ft Mcfry of pofitiun ptrpendicuUiri to Ike axis of the ab- cuMT, and rtciprocaiU, iterpendicvtars to the axis of the abcttMM mrr tmngtnaneM of thit form. For, putting •{- a and ^a for straight lines lying in opporite directions, according in ^ Fran^ ais, the (luantity =t: av'— 1 is a mean pruiwrtional of magnitude and position between Now it bm bean prtaaiscti, that a line, which

peparty baleags alaa to i<m aamerhy theimagiaaey n Jiirfmi My. iriib aU dia

ia a aman proportianal in nrngnitude and position be- tween two Kalas, oiwht to biwct the angle they con« laai ; Iherefcrt. in Uic ptasnt case, the mean must be patpaaJitalar la the ana of the abciss*. and will lie sbo^ or below tha azia according as it ik +<iv/ 1, or «%/ I. Reupracalty, everv perp<-ndiculsr to the axis of tha abdsa* matt, aeeorcung to the same princi>

nmaHi^ he Mtad af the Ihanae we imM liwiifti thai

be in

^^nV ^^^^ ^BP^ ^^^^M^^^^nB>

9mmt rvary «*• resdria

laa oms^ mmmy i Wee ■iattiely tha

wH bt la tha avMsnoe of o^ ia pravad af tha by* It* held liar of af tha aarvai^ wiU d^grsas^ a aaaas ia maat aat by aaaipaai Of lala yaata ilhm threey «

ImabseaWa^bkfitvatdlMM .

#>AW. TVaa* IHUf. aim by M. Argaadt hi a thw titla^ Xasai sar aa

MldtelWaia IMC, and again by M J. f\ lUa9da la Hmmmaapnmtip—iegu mtuir 4t fB\i»ma s* mli rpit aaUsa gmamfriaae aas tjKt^mM san^j^saasras^ pabnsiMi in Ammidm 4r UmKtmm^m, >apl. lti«. Aecordbig to

aian tf asramdfc^ania j m i^a tha ^^ma^ew'caiv^ ••—I,

aen .fa and —a ; it ia of the fiinn dtdv/— 1. of M. Fran^ais' demon »tration ; to ba by no miwn* utiafactory. Wa d, thM in seeking the mean ofroa*- batvaca -fa and o. he would tha maaa of maaalladt mdrpmdently of and then tiw bmbb or paailioB inilctK-mlrntly af magaitada Tbaat woald aava raquireti different oMiauaat; tha fcai would have aivaa y^a x ''=<> for tae aMaa af la^giiiHidi . and, pottnm f lor a right an* git, tha sacosd woald have givan \{n-\-q)—q for the amaa af poaiiian. Wa caaaol, however, >ce any useful from tha renilL The author of tha Ham 4-a and a, teems to at aaea wfch myiituJa and nom. a timt ba aaeCs tha maan by a to them M things having only

iha thaaty, by crilisy I hava iavailad thaai al

_^ f writer*, re*

BiimnHal gaumeoically hy a asiyjibiaha to a otraight line « thai fc. if a baa taihe rmbt ba laaiMiJ by ^ a. aadaaa^ttnato tha Mk by a. dMaathMNna piiymdi«atw fa and cqaal «a either of thaw will bt 1by:l=aV'— I'- ravMaalhii.thalaNi

by Mr. Buee in support of the Irath of tha prapoattian, that y^ I is the sign of per* peadicularity. is not more condusivc. He nuppaem thrsa equal straight line* to meet in s point, two of them to be in one straight line, and the third to be at right angle* to them biith. He calls the line taken to the right 4- 1 , then that taken to the left, be mys. mutt be T ; and the third, which must be a mean propor« tional bctwean tfaaai, most be ^—{V), at more sim« ply v/— I. Hence he infers, that v'— I is the sign of ftrftudicmlariiif. The inoondasivencss of this reason- ing, hm been well exposed by an able critic in the Edfnbargh Review, vol. ail lulv I80H, where it is ob- ssrved. that any imaginable conclusion might have been derived in the same manner. For example, the third Ifa^ iHUad of batng at riglit angkt, may be nippotad

J %t X 5