:

BOOK XI. be made to coincide; for if the parallelogram CH be applied to GE, fo that B be on F, and BC on FG, the triangles BCF, CGF fall on different fides of the common plane, GE and the fame is the case when B is applied to A, and BC to AE; for if the parallelograms be thus made to coincide, the triangles BCF, ADE fall on different fides of the common plane GE. In order to prove the prifms to be equal by Prop. B, it is neceffary that the angle CBH be equal to the angle AGF, and FEH to CGA, which can never happen, except each of them be a right angle; for AGF, GFE, that is, AGF, CBH are together equal to two right angles, and therefore, if one of them be lefs, the other must be greater than a right angle: Wherefore this propofition is not fufficiently demonftrated in the former editions. A new demontration is now given by the help of Prop. C, by which means we are enabled to compare together all folids, though contained by planes not alike fituated in respect of one another.

PROP. XXIX. XXXI. XXXIII.

A MORE general demonftration is given of the 29th: and the figures of the 31st and 33d are made more fimple, by means of which the demonftration of the latter is fhortened. Profeffor Play fair, in his demonstration of this 33d, fuppofes, that fimilar parallelopipeds have to one another the ratio which is compounded of the ratios of their bafes, and of their infifting lines; but as he has not before proved, that the infifling lines are to one another as the altitudes, the fuppofition is unwarrantable.

THE demonftration of this is now much fhortened, by making the conftruction of the whole fimilar to what was before given of the fecond part of the firft cafe; and by demonstrating the two parts of the propofition separately.

THE only ufe of this propofition in the Elements is, to demonftrate the following corollary by it; and the corollary is. ufed in the following propofition. But it is beft to demonstrate the corollary firft, because it is the only part that is ufed, and its demonftration is fhorter than the other; and that of the propo-, fition may be eafily made from it. This method is now follow-. ed, and the demonftration is made shorter by the help of Prop. A.

BOOK

N the enunciation of the fecond propofition, it is fuppofed Book XII. poffible for fome ftraight line to be equal to the circumference of a circle; and if the axiom be granted, this is also evident; for it follows from the axiom, that the perimeter of any circumfcribed polygon is greater than the circumference, and that the perimeter of any infcribed polygon is lefs than it; therefore there must be fome ftraight line lefs than the former, and greater than the latter, that is neither greater nor less than the circumference, and confequently is equal to it.

It is likewife faid, in the conftruction of this 2d, that if DF be greater than the circle, there is fome rectangle DH equal to the circle. It would not alter the demonstration, though DH be taken greater than the circle; and this is fhown to be poffible in the firft Prop. For if EF be bifected by a straight line parallel to ED, it will cut off the half of the rectangle DF; and in the fame manner may the half of the remainder be cut off; and fo on; therefore, if this be done continually, there fhall at length remain a rectangle lefs than the excess of DF, above the circle: let this be the rectangle HG; therefore the rectangle DH is greater than the circle. But it is fuppofed, in the propofition, that there is fome rectangle equal to the circle; and in the demonftration it is proved, that it can neither be greater nor lefs than DF; therefore DH may be taken any rectangle whatever lefs than DF; and then the import of the first part of the demonftration is to inquire, what would follow from fuppofing it equal to the circle: and the abfurd consequence which we neceffarily draw from it, obliges us to relinquish the fuppofition, that any rectangle less than DF is equal to the circle, or that DF is greater than the circle. So that it is not neceffary to accuracy, that we be able to determine the magnitude of DH, be a construction; it is fufficient if its being equal to the circle, be a fuppofition that is evidently poffible. And the like is to be obferved in the following propofitions. In indirect demonftrations, actual conftructions may be often fpared, in cafes where they are neceffary in direct ones. Thus, in the 6th of the ift Book, it evidently follows, from our fuppofing AB greater than AC, that fome part of AB is equal to AC; and this part, whatever be its magnitude in refpect of AB, may be called BD. And it seems to be ufelefs labour to attempt the construction of a line or figure which we intend to prove impoffible.

IN

LOGAR.

IN Trigonometry, the principles are explained fully, and the

nature and use of the Tables of Natural Sines, Tangents, and Secants are fhewn, as well as their conftruction. But as it is nfual to perform the operations by Logarithms, their nature fhall be explained here.

LOGARITHMS.

A SERIES of numbers, which increase by the fame difference, is called an Arithmetical Progreffion and a feries of numbers, which increase by a common multiplier, is called a Geometrical Progreffion.

Let the first term of an arithmetical progreffion be placed above the first of a geometrical one, and the rest in order, thus:

Arithmetical progreffion,
Geometrical progreflion,

0, 1, 2, 3, 4, 5, 6, &c. 1, 4, 16, 64, 256, 1024, 4096, &c.

Then it is evident, that if we are to multiply any term of the geometrical ferics, as 64, by another term, as 16, the product can be found by adding the correfponding terms of the arithmetical feries, viz. 2 and 3, for the fum 5 is the term correfponding to the product 1024: Thus, the ufe of a Table of arithmetical progreffionals correfponding to a geometrical progreffion, is evident; but in order to this, the terms must be increased in both, fo that the geometrical may contain all the natural numbers of arithmetic. This may be done, by finding geometrical mean proportionals between the terms of the geometrical progreffion, and as many arithmetical means between the terms of the arithmetical one. Now, an arithmetical mean is half the fum of the adjacent terms, and a geometrical mean is the fquareroot of the product of the adjacent terms. If this be done, the feries will contain twice as many terms as before, and will become

Arithm. prog. o, 1, 1, 11, 2, 21, 3, 34, 4. 4. 5, 51, 6, &c.
Geom. prog. 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, &c.

Thefe progreffions may be interpolated in the fame manner, by new terms; and the operation may be continued until all the natural numbers occur in the geometrical progreffion. And then the terms of the arithmetical feries correfponding to these numbers, are called the Logarithms of them.

Hence, Logarithms are artificial numbers, by the affiftance of which, addition fupplies the place of multiplication, and fubtraction of divifion.

In the common tables of logarithms, the progreffions are LOGAR. Arithmetical. o, I, 2, 3, 4, 5, &c. Geometrical. 1, 10, 100, 1000, 10,000, 100,000, &c. having terms interpolated continually, as was fhewn in the former feries, until all the natural numbers are in the geometrical feries, and the numbers in the table of logarithms are those of the arithmetical feries, which correfpond with the natural numbers in the geometrical feries.

Hence, the logarithms of all numbers between 1 and 10, are fractions; the logarithms of all numbers between 10 and 100, are mixed numbers that have 1 for the integer, and those of numbers between 100 and 1000, have 2 for the integer; and so on: That is, the units in the integer are always lefs, by one, than the places in the correfponding number. The integer is called the Index, because it fhews the number of figures in the anfwering number.

TO find the Logarithm of any Number from the
Tables.

Look for the three highest figures in the margin on the left fide, and in that line, in the column which has the fourth figure at the top, you will find the logarithm for thefe four figures. If the number have more figures in it, take the difference between this logarithm, and the next greater, and multiply it by the remaining figures, and from the product cut off as many figures as are in the multiplier, the reft added to the logarithm for the first four figures, give the logarithm required. The index is always one lefs than the number of integers.

TO find the Number for a given Logarithm.

IF the logarithm be found in the table, the three first figures are on the fame line in the margin, and the fourth is at the top of the column. But if the logarithm be not in the table, take the number anfwering the next lefs logarithm, and subtract this lefs logarithm from the given one, and alfo from the next greater; then annexing cyphers to the first remainder, divide it by the other, to get the fifth, fixth, &c. figures. The integers mu be one more than the index, and the reft are decimals.

Thus, To find the logarithm of 73284, on the line that has 732 in the margin, and in the column that has 8 at the top, there is found 854985, the logarithm of 7318, and the diffe rence between it and the next logarithm is 60, which, multi

plied

LOGAR. plied by 4, gives 240; therefore, adding 24 to .864985, we have 4.865009 for the logarithm of 73284, with 4 for an index, because the number has five places.

Again, To find the number anfwering to the logarithm 4.597179. Look for the next less logarithm, and the number anfwering to it is 3955, and the difference between it and the given logarithm is 33, and the difference between it and the next greater in the table is 110; therefore divide 330 by 110, and the quotient 3, annexed to 3955, gives 39553 for the number anfwering the given logarithm.

Of the Tables of Sines, Tangents, and Secants.

THESE tables have the degrees at the top, and the minutes on the left fide of the page, when the degrees are not above 45°; but if they be greater than 45°, the degrees are at the foot, and the minutes on the right fide.

If the logarithms of the numbers in these tables be taken, and arranged in the fame manner in tables, they form the tables of Artificial Sines, Tangents, and Secants, which supply the place of the natural ones, in the fame manner that logarithms fupply the place of natural numbers.

TO find the Artificial Sine of 37° 23′ 12′′.

Look for the page that has 37° at the top, and there, on the line that has 23' on the left fide, and in the column titled fine, is 9.7832922, the fine of 37° 23′: and the difference between it and that of 37° 24', is 1653. Then, as 60" to 12", fo is 1653 to 331, the proportional difference for 12", which, added to 9.7832922, gives 9.7833253 for the fine of 37° 23′ 12′′.

Again, To find the degrees, and parts of a degree, of which 10.2738462 is the artificial tangent.

Look for the nearest tangent, 10.2737163, and because it is titled Tang. at the foot, take the degrees at the foot, and the minutes on the right, and it gives 61° 58′. And the difference between this tangent and the one above it is 3046, and the difference between it and the given one is 1299; therefore, as 3046 to 1299, fo is Co" to 26"; fo that 10.2738462 is the tangent of 61o, 58′, 26′′.

Te