LOG
LOG-
^erboiic tpaces are • the parallelogram of the hyperbola, as the lines D G, F E arc to the fubtangent F O. So that if this parallelogram be fuppofed = 0.4342944819, every hyperbolic fpace comprized between two ordinates to one of the afymptotcs, will be to this parallelogram, as the loga- rithm of the ratio of the two ordinates, or the difference of the logarithms of the numbers expreffing the proportions of thofe ordinates, is to 0.4342944819; taking the loga- rithm of 10 figures befides the characteristic. "We have mentioned one method of finding the fubtangent of a given logarithmic from Huygens in n° 4. But this may be othcrwife done thus, produce any ordinate CD from
C to E, fo that the ratio of C E to C D be equal to the
ratio modularis : and the right line E B drawn from E paral- lel to the afymptote, and meeting the curve in B, will be equal to the fub-tangent. The ratio nodularis is 2.7182818, tsV. to 1, or 1 to 0.3678794, &c. for which may be fubfti- tuted the approximations 11 to 4, 87 to 32, &c, greater than the true, and 8 to 3, 19 toy, 1061039, & Cm ^ s t ' lan the true ratio. See Cotes, Harm. Menf. p. 7 and 16. While the bale O P increafes uniformly, let the ordinate P^i increafc or decreafe proportionally, as mentioned under thi head Logarithm ; that is, let the velocity of p in the tli rcction Os, be always as the ordinate Vp ; then will the pointy defcribe the logarithmic curve. The Safe O P is always
O P AT DTE
the logarithm of the ordinate P p ; and if the uniform motion of P be equal to the motion of p at in the direction O 0, then is the ordinate O the modulus of this fyftem of loga- rithms ; and the fluxion of any ordinate is to the fluxion of the bafe as that ordinate is to O 0.
By the general property of tangents, the fluxion of any ordi- nate is to the fluxion of the bafe, as that ordinate is to the fub-tangent. Hence the fub-tangent O T, or AT, will be equal to O 0. The fub-tangent will therefore be con- ftant, In any other logarithmic curve, the fub-tangent will always be equal to that ordinate, whofe fluxion is equal to the fluxion of the bafe, that is, the modulus always ex- prefles the fub-tangent of the logarithmic curve. See the ar- ticle Logarithm.
The fub-tangent of the logarithmic exprefling the moduli of that fyfl-em of logarithms, the fubtangent will be the logarithm of the ratio modularise that is, of the ratio of two ordinates raifed at the extremities of the fub-
- tangent j and as the ratio modularh is the fame in all fyf. terns of logarithms, fo the ratio of the two ordinates placed at the extremities of their refpective fub-tangents, is the fame in all logarithmic curves whatfoever. When the logarithmic curve is fuppofed to be defcribed ex- ponential quantities are eafdy determined by it. Let the or- dinate A a, be expreffed by A; let OP be to O A as any quantity exprefTed by x is to 1 : then the ordinate Pp may be expreffed by A x , an exponential quantity of the firft de- gree when x is variable. See Exponential, Cycl.
Tc reftify the logarithmic Curve. Let it be required to find the length of the logarithmic arc E e. Draw the perpendi-
larsELA, ela, to the afymptote, and having drawn the tangents EF, tf 3 take AL equal to the excefs of the tan-
gent above the fub-tangent A F ; and a I, equaJ to the excefs of the tangent ef above af; then having drawn LM, lm parallel to the afymptote, if the difference of the tangents EF— */", be added to the difference of the parallels lm — L M, the aggregate will be equal to the arc E e. Cotes, Harm. Menf. p. 23, 24.
LOGISTIC (Cycl.) — Logistica, in geometry, is ufed to %- nify the logarithmic curve. Huygens introduced this term in his treatife of the caufe of gravity, at the end of which lie gives us feveral curious properties of this curve. See Logarithmic curve.
LOGOGRAPHI, Ao^oy^aipo,, among the antients, were the fecretaries of the logiftae, and kept an account of the public revenues. See Logist^, Cycl.
LOGOTHETA, an officer under the emperors of the eair, who kept an account of the various branches of public and private expence.
There were feveral kinds of them, diftingmihed by the par- ■ ticular branch they fuperinteiided, as the logotheta m ty<y*a, or poft-mafl:er general, logotheta tm oixiwxm, or matter of the houfhold, &c Vid. du Cange, GlofT. Lat. T. 2. in voc.
LOLIUM, Damell, in the Linnsean fyftem of botany, makes a diftinit. genus of plants. The diitinguiftiing characters of which are, that the calyx is a glume containing feveral flowers, univalve, oblong, ftrait, pointed, and very rigid, containing the flowers gathered in a fort of fpike placed clofely acainif. the ftalk. The flower is compofed of two valves ; the in- ferior is narrow, and pointed, rolled up, and of the length of the cup; the upper valve is fhorter, ftrait, obtufe, and hol- low in its upper part. The ftamina are three filaments, very flender, and fhorter than the cup ; the anthers are oblong. The germen of the pifHllum is of a turbinated form, the- ftylcs are two in number, capillary, and reflex, and the ftio-- mata are plumofe. The flower clofely furrounds the feed, and at a proper time opens, and lets it fall. The (eed is fingle, oblong, of a compreffed fiiape, and convex on one fide, and flat and furrowed on the other, Liniiai Gen. Plant, p. 16.
The antients ufe the word lolium, as the name of a difeafed ftate of corn of any kind, and have given accounts of fe- veral forts of difeafed appearances, to all of which they indif- ferently gave this name. When the corn was deftroyed by the great and unnatural increafe of the root, they called it polypode lolium. This never produced good ears, but often a great number of itraggling-and ufelefs branches. Another kind of lolium was owing to the feeds in the car being Hinted in growth, never arriving at their natural fize, and becom- ing of a greyifh colour. Another kind was, when the feeds in the ear were larger and harder than they naturally ought to be, and of a black colour ; and another kind was, when the grains grew large, and oblong, and were crooked, and marked with a white fpot in the middle, but black in all the other parts. The Germans are fubjccl to this diftemperature of their grain, and they call it mutter horn. Thefe diftempered ftates of the corn, when they affeiSt a great quantity of it in the fields, and are gathered and ground along with the natural and found corn, for the ufe of the common people, are the caufe of epilepfics, and many Other terrible difeafes raging violently among them, and carrying off great numbers.
The antients were well acquainted with the pernicious na- ture of this fort of corn ; and Ariftotle, Theophraftus, Pliny, and many others, the moft eminent hiftorians, na- turalifts, and poets, give accounts of difeafes brought on by- eating it. The caufing diftempers of the eyes has been at- tributed to this, even to a proverb ; a man being faid Islia viclitare, to feed on lolium, or this diftempered corn ; iu- ftead of, 1 in plain words, being blind : but though thus hurt- ful in food, this lolium was ufed in medicine, in cafes of pains in the head and other parts, and in inflammations and ulcers; in thefe cafes it was ufed externally, and inhyfteric complaints a few grains of it were often given internally.
LOMBAIRE externe, in anatomy, a name given by Win- flow, and fome other of the French writers, to a mufcle, called by the earlier authors the primus dorji, and now ge- nerally known by the name quadrants lumborum. The French call this the external lumbal mufcle, in diftin<ftioii from the pjlas ??iagnus, which they call the kmbalre interns.
Lombairk interne, in anatomy, a name given by Window, and fome other French writers, to the mufcle generally called pfoas magnus.
LOMENTUM, a word ufed by the old writers on medi- cine, to exprefs a meal made of beans, or bread made of this meal, and ufed as a warn. Sec Detersorium. Others have applied it to the French chalk, or morochthus, ufed by the fcowerers of cloaths, which is brought over in large cakes, refcmbling loaves or cakes of bread.
LOMWIA, in zoology, the name of a web footed water fowl common on the Englilh fhores, and called in different places the guiUem, guillemot, fea ben, kiddoiv, and jkout : the laft name, however, is fomewhat equivocal-, as the
Scotc h call the common razor hill by this name. The lomtuia, or kiddow, is much like the raaor bill, but
