LOG
(4<fe)
LOG
tor, its Logarithm muft be the Difference of the Logarithms of thofetwo; fo that the Numerator being fubftracted from the Denominator, the Difference becomes negative. Stfelius obferved, that th& Logarithms of a proper Frac- tion muft always be negative, if that of Unity be o 5 which is evident, a Fraction being lefs than one.
For an improper Fraction, v.g. A, itsNumerator being greater than its Denominator, its Logarithm is had, by fub- ftra&ing the Logarithm of the latter from that of the former.
The Logarithm of 5=0.9542425 Logarithm of 5=0.5989700
Logarithm .'=0.2553725
In the fame manner may a Logarithm of a mixed Num- ber, as 3 f be found, it being firft reduced into an im- proper Fraction o *y.
To fnd the Number cor refunding to a Logarithm, greater than any in the Table : Firft {torn the given Logarithm, fub* ftract the Logarithm of 10, or ico, or 1000, or 10000, till you have a Logarithm that will come within the Compafs of the Table ; find the Number correfponding to this, and multiply it by 10, or ico, or 1000, or ioooo, the Product is the Number required.
Suppofe, for Inftance, the Number correfponding to the Logarithm 7.7589982 be required ; fubftract the Loga- rithm of the Number 10000, which is 4.0000000 from "7.7589982, the Remainder is 3.7589982, the Number correfponding to which is 5741 ^g, this multiplied by iccoo, the Product is 5 741 r 100, the Number required.
To fnd the Number correfponding to a negative Logarithm. To the given negative Logarithm, add the I aft Logarithm of *he Table, or that of the Number iooqo 5 i.e. fub- ftract the firft from the fecond, and find the Number correfponding to the Remainder 5 this will be the Nume- rator of the Fraction, whofe Denominator will be 10000 5 v. g. fuppofe it be required to find the Fraction correfpond- ing to the negative Logarithm o.^6j9']6y, fubftract this from 4-icocooo
The Remainder is — 3.^320233, the Number correfponding to which is 4285^, the Fraction fought therefore is itg£5£ Tft e Reafon of the Rule is, that as a Fraction is the Quotient arifing on the Divifion of the Numerator by the Denominator, Unity will be to the Fraction as the Denominator to the Numerator j but as Unity is to the Fraction correfponding to the given ne- gative Logarithm, fo is 10000 to the Number correspond- ing to the Remainder; therefore if 10000 be taken for the Denominator, *he Number will be the Numerator of the Fraction required.
. To find a fourth Proportional to three given Numbers. Add the Logarithm of the fecond to that of the third, and from the Sum fubftrac'r the Logarithm of the firft, the Re- mainder is the Logarithm of the fourth required. E. g. let the given Number be 4. <S8. and 3,
Logarithm 68=1.8325089 Logarithm 3=0.4771213
Sum=2. 3096302 Logarithm 4=o.<5"o20tfoo
Logarithm required
1.7075702
The Number in the Tables correfponding to which is 51. This Problem is of the utmoft XJk in Trigonome- try. See Trigonometry.
LOGIC is the Art of Thinking juftly 5 or of making a right Ufe of our rational Faculties in defining, dividing, and reafoning. The word is derived from hiy&, Sermo, Difcourfe; Thinking being no more than an inward, mental Difcourfe, wherein the Mind converfes with itfelf. Logic is fometimes alfo call'd DialeBics, from JW^fe/, to rea- fon ; and fometimes the Canonical Art, as being a Canon or Rule for directing us in our Reafonings. As, in order to think aright, 'tis neceffary that we apprehend, judge, difcourfe, and difpofe or methodize rightly : hence Ap- preheniion, Judgment, Difcourfe, and Method become the four fundamental Articles of this Art 5 and 'tis from ourRefledtions on thofe Operations of the Mind, that Lo- gic is, orought to be wholly drawn. My Lord Bacon di- vides Logic into four Branches, according to the Ends pro- pofedineach: fora Man reafons either to find what he feeks, or to judge of what he finds, or to retain what he judges, or to teach what he retains ; whence arife fo many Arts of Reafoning, viz. the Art of Inquifition or Inven- tion, the Art of Examining or Judgment, the Art of Pre- ferring or of Memory,and the Art of Elocution or Deliver-
ing, which fee. Logic having been extremely abufed, i* now in a general Difrcpute. The Schools have fo clogged it with barbarous Terms and Phrafes, and have run it out fo much into dry ufelefa Subtilties, that it feems rather in- tended to exercife the Mind in Wrangling and Difputation, than to affift it in thinking juflly. 'Tis true, in its Origi- nal, it was rather intended as the Art of Cavilling than of Reafoning ; the Greeks, among whom it had its Rife, being a People who piqued themfelves mightily upon their being able to talk ex tempore, and to argue by Turns on either Side the Queftion. Hence their Dialectic i, to be always furniihed with Arms for fuch Rencounters, in- vented a Set of Words and Terms, rather than Rules and Reafons, fitted for the Ufes of Contention and Difpute. Logic, then, was only an Art of Words, which frequently had no Meaning, but ferved well to hide Ignorance, in-' ftead of improving Knowledge, to baffle Reafon inftead of aflifting it, and to confound the Truth inftead of clear-* ing it. All that Heap of Words, which we have bor- rowed from the old Logic, is of little Ufe in Life, and is fo far outof the common Ufage ; that the Mind does not attend to them without Trouble, and finding nothing in them to reward its Attention, foon difcharges " itfelf, and lofcsall Ideas it had conceived of them : But Logic dif- engaged from the Jargon of the Schools, and reduced in- to a clear and intelligible Method, is the Art of conduct- ing the Reafon in the Knowledge of Things, and the Dif- covery of Truth. From its proper Ufe we gain feveral very confiderable Advantages ; for, (i.)TheConfideration of Rules incites the Mind to a clofer Attention and A ppli- cation in Thinking, fo that we hereby become affured that we make the beft Ufe of our Faculties. (2.) We hereby more eafily and accurately difcover and point out the Errors and Defects in our Reafonings 5 for the common Lightof Nature, unaftlfted by Logic, frequently obferves an Argumentation faulty, without being able to deter- mine wherein the precife Failure confifts. (3.) By thefe Reflections on the Order and Manner of the Opera- tions of the Mind, we are brought to a more juft and. compleat Knowledge of the Nature of our own Under* ftanding.
LOGISTA, the Title of a Magiftrate at Athens, whofe Bufinefswas to receive and pafs the Accounts of Officers upon their laying down their Pofts, The Logiftce were in Number ten.
LOGISTIC, or Logarithmic Line, a Curve fo called, from its Properties and Ufes in conftructing and explain- ing the Nature of Logarithms. If the right Line A X, (Tab. Analyfis, Fig. 12.) be divided into any Number of equal Parts, and to the Points of thofe Divisions A P p t Sec. be drawn Lines continually proportional, the Points N M m, &c. form the Logifiic Curve.
Cor. r. The Abfciffes A P, A f>, &c. are the Logarithms of the Semiordinatcs P M, p m, &c.
Hence if A P = x, A f =s % P M —y, p m = z , and their Logarithms y and 2.= / y and I z, * will be *stiy 9 and v = l z, confequently x : v ■==. I y : I z, that is the De- nominators of the Ratio's A N, P M, and A N/> m, are to one another as the Abfciffes A P and A p.
Cor. 2. Hence it follows, that there may be infinite 0- therLogiftic Lines invented, provided xmivm; : I y ; I z, that any of the Roots or Powers may be the Logarithms of the Semiordinates.
Cor. 3. The Logiftic will never concur with the Axis except at an infinite Diftance, fo that A X is its Afymp- tote.
LOGISTIC, or Logarithmic Spiral, a Line, whofe Conftruction is as follows : Divide the Quadrant of a Cir- cle into any Number of equal Parts in the Points P, p p, &c. (Tab. Analyfis, Fig. 11.) and from the Radii CP, Cf, Cp, &c. cutting off CM, Cm, Cm, &c. conti- nually proportional, the Points M m, m, &c, form the Lq- gijiic Spiral.
Cor. The Arches therefore A P, A p, &c. are the Logarithms of the Ordinates CM, Cm, ckc. whence alfo it follows, that there may be infinite Logifiic Spi- rals;
LOGISTICAL ARITHMETIC, was formerly the Arithmetic of Sexagefimal Fractions, ufed by Aftrono- mers in their Calculations, ltwasfo called from a Greek Treatife of one Barlaamus Mofuetias, who wrote about Sex- agefimal Multiplication very accurately, and entitulcd his Book A8?(57Ktj. This Author, Vofftus places about the Year 1350, but mi flakes the Work for a Treatife of Algebra. Thus alfo Sbafarty, mTahala Britannic*, hath a Table of Logarithms adapted to Sexagefimal Fractions, which therefore he calls Logijiical Logarithms ; and the expeditious Arithmetic of them, which is by this means obtained,, and by which all the Trouble of Multipli- cation and Divifion is faved, he calls Logijiical Aritb~ mctic though fonae by Logifiics will underftand the Dddddd firil
