are the indexes or exponents (mostly whole numbers and decimal fractions, consisting of seven places of figures at least) of the powers or roots (chiefly broken) of a given number; yet such indexes or exponents, that the several powers or roots they express are the natural numbers 1, 2, 3, 4, 5, &c. to 10 or 100000, &c. (as, if the given number be 10, and its index be assumed 1000000, then the root of 10, which is 1, will be the logarithm of 1; the root of 10, which is 2, will be the logarithm of 2; the root of 10, which is 3, will be the logarithm of 3; the root of 10, which is 4, will be the logarithm of 4; the root of 10, which is 5, will be the logarithm of 5; &c.) being chiefly contrived for ease and expedition in performing of arithmetical operations in large numbers, and in trigonometrical calculations; but they have likewise been found of extensive service in the higher geometry, particularly in the method of fluxions. They are generally founded on this consideration, that if there be any row of geometrical proportional numbers, as 1, 2, 4, 8, 16, 32, 64, 128, 256, &c. or 1, 10, 100, 1000, 10000, &c. and as many arithmetical progressional numbers adapted to them, or set over them, beginning with 0,

\[ \begin{align*} &\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \\ &\{1, 2, 4, 8, 16, 32, 64, 128, 256, \ldots\} \end{align*} \]

then will the sum of any two of these arithmetical progressions added together, be that arithmetical progression which answers to or stands over the geometrical progression, which is the product of the two geometrical progressions over which the two assumed arithmetical progressions stand: again, if those arithmetical progressions be subtracted from each other, the remainder will be the arithmetical progression standing over that geometrical progression which is the quotient of the division of the two geometrical progressions belonging to the two first assumed arithmetical progressions; and the double, triple, &c. of any one of the arithmetical progressions, will be the arithmetical progression standing over the square, cube, &c. of that geometrical progression which the assumed arithmetical progression stands over, as well as the \( \frac{1}{2}, \frac{1}{3}, \ldots \) of that arithmetical progression will be the geometrical progression answering to the square root, cube root, &c. of the arithmetical progression over it; and from hence arises the following common, tho' lame and imperfect definition of logarithms, viz.

"That they are so many arithmetical progressions, answering to the same number of geometrical ones."

Whereas, if any one looks into the tables of logarithms, he will find, that these do not all run on in an arithmetical progression, nor the numbers they answer to in a geometrical one; these last being themselves arithmetical progressions. Dr Wallis, in his History of Algebra, calls logarithms the indexes of the ratios of numbers to one another. Dr Halley, in the Philosophical Transactions, no 216, says, they are the exponents of the ratios of unity to numbers. So also Mr Cotes, in his Harmonia Mensurarum, says, they are the numerical measures of ratios. But all these definitions convey but a very confused notion of logarithms. Mr Macaulay, in his Treatise of Fluxions, has explained the nature and genesis of logarithms agreeably to the notion of their first inventor Lord Napier. Logarithms then, and the quantities to which they correspond, may be supposed to be generated by the motion of a point; and if this point moves over equal spaces in equal times, the line described by it increases equally.

Again a line decreases proportionally, when the point that moves over it describes such parts in equal times as are always in the same constant ratio to the lines from which they are subducted, or to the distances of that point, at the beginning of those lines, from a given term in that line. In like manner, a line may increase proportionally, if in equal times the moving point describes spaces proportional to its distances from a certain term at the beginning of each time. Thus, in the first case, let ac be to ad, cd to ce, de to df, ef to fg, &c., always in the same ratio of QR to QS; &c. &c. &c., suppose the point P sets out from a, describing ac, cd, de, ef, fg, in equal parts of the time; and let the space described by P in any given time be always in the same ratio to the distance of P from sat the beginning of that time; then will the right line ac decrease proportionally.

In like manner, the line ac, (ibid. no 3.) increases proportionally, if the point p, in equal times, describes the spaces ac, cd, de, ef, &c. so that ac is to ad, cd to ce, de to df, &c. in a constant ratio. If we now suppose a point P describing the line AG (ibid. no 4.) with an uniform motion, while the point p describes a line increasing or decreasing proportionally, the line AP, described by P, with this uniform motion, in the same time that ac, by increasing or decreasing proportionally, becomes equal to ap, is the logarithm of op. Thus AC, AD, AE, &c. are the logarithms of Logarithms of \( ac, cd, oe, \) &c., respectively; and \( ac \) is the quantity whose logarithm is supposed equal to nothing.

We have here abstracted from numbers, that the doctrine may be the more general; but it is plain, that if AC, AD, AE, &c. be supposed 1, 2, 3, &c. in arithmetic progression; \( ac, cd, oe, \) &c. will be in geometric progression; and that the logarithm of \( ac, \) which may be taken for unity, is nothing.

Lord Naper, in his first scheme of logarithms, supposes, that while \( ep \) increases or decreases proportionally, the uniform motion of the point \( P, \) by which the logarithm of \( ep \) is generated, is equal to the velocity of \( p \) at \( a; \) that is, at the term of time when the logarithms begin to be generated. Hence logarithms, formed after this model, are called Naper's Logarithms, and sometimes Natural Logarithms.

When a ratio is given, the point \( p \) describes the difference of the terms of the ratio in the same time. When a ratio is duplicate of another ratio, the point \( p \) describes the difference of the terms in a double time. When a ratio is triplicate of another, it describes the difference of the terms in a triple time; and so on. Also, when a ratio is compounded of two or more ratios, the point \( p \) describes the difference of the terms of that ratio in a time equal to the sum of the times in which it describes the differences of the terms of the simple ratios of which it is compounded. And what is here said of the times of the motion of \( p \) when \( ep \) increases proportionally, is to be applied to the spaces described by \( P, \) in those times, with its uniform motion.

Hence the chief properties of logarithms are deduced. They are the measures of ratios. The excess of the logarithm of the antecedent above the logarithm of the consequent, measures the ratio of those terms. The measure of the ratio of a greater quantity to a lesser is positive; as this ratio, compounded with any other ratio, increases it. The ratio of equality, compounded with any other ratio, neither increases nor diminishes it; and its measure is nothing. The measure of the ratio of a lesser quantity to a greater is negative; as this ratio, compounded with any other ratio, diminishes it. The ratio of any quantity A to unity, compounded with the ratio of unity to A, produces the ratio of A to A, or the ratio of equality; and the measures of those two ratios destroy each other when added together; so that when the one is considered as positive, the other is to be considered as negative. By supposing the logarithms of quantities greater than \( ac \) (which is supposed to represent unity) to be positive, and the logarithms of quantities less than it to be negative, the same rules serve for the operations by logarithms, whether the quantities be greater or less than \( ac. \) When \( ep \) increases proportionally, the motion of \( p \) is perpetually accelerated; for the spaces \( ac, cd, de, \) &c. that are described by it in any equal times that continually succeed after each other, perpetually increase in the same proportion as \( ep \) decreases.

If the velocity of the point \( p \) be always as the distance \( ep, \) then will this line increase or decrease in the manner supposed by Lord Naper; and the velocity of the point \( p \) being the fluxion of the line \( ep, \) will always vary in the same ratio as this quantity itself. Logarithms, we presume, will give a clear idea of the genesis or nature of logarithms; but for more of this doctrine, see Maclaurin's Fluxions.

Construction of Logarithms.

The first makers of logarithms had in this a very laborious and difficult task to perform. They first made choice of their scale or system of logarithms, that is, what set of arithmetical progressions should answer to such a set of geometrical ones, for this is entirely arbitrary; and they chose the decuple geometrical progressions, 1, 10, 100, 1000, 10000, &c. and the arithmetical one, 1, 2, 3, 4, &c. or 0, 1, 2, 3, 4, &c. as the most convenient. After this they were to get the logarithms of all the intermediate numbers between 1 and 10, 10 and 100, 100 and 1000, 1000 and 10000, &c. But first of all they were to get the logarithms of the prime numbers 3, 5, 7, 11, 13, 17, 19, 23, &c. and when these were once had, it was easy to get those of the compound numbers made up of the prime ones, by the addition or subtraction of their logarithms.

In order to this, they found a mean proportion between 1 and 10, and its logarithm will be \( \frac{1}{2} \) of that of 10; and so forth, they found a mean proportional between the number first found and unity, which mean will be nearer to one than that before, and its logarithm will be \( \frac{1}{2} \) of the former logarithm, or \( \frac{1}{2^2} \) of that of 10; and having in this manner continually found a mean proportional between the last mean, and bisected the logarithm, they at length, after finding 54 such means, came to a number 1.000000000000000127819149320032342, so near to 1 as not to differ from it so much as 1 in \( 10^{100} \); and they divided this part, and found its logarithm to be 0.0000000000005551115123125782702, and 0.00000000000012781914932003235 to be the difference whereby 1 exceeds the number of roots or mean proportions found by extraction; and then, by means of these numbers, they found the logarithms of any other numbers whatsoever; and that after the following manner: Between a given number whose logarithm is wanted, and 1, they found a mean proportional, as above, until at length a number (mixed) be found, such a small matter above 1, as to have 1 and 5 cyphers after it, which are followed by the same number of significant figures; then they said, As the last number mentioned above is to the mean proportional thus found, so is the logarithm above, viz. 0.0000000000005551115123125782702, to the logarithm of the mean proportional number, such a small matter exceeding 1 as but now mentioned; and this logarithm being as often doubled as the number of mean proportions (formed to get that number) will be the logarithm of the given number. And this was the method Mr. Briggs took to make the logarithms. But if they are to be made to only seven places of figures, which are enough for common use, they had only occasion to find 25 mean proportions, or, which is the same thing, to extract the root of 10. Now having the logarithms of 3, 5, and 7, they easily got those of 2, 4, 6, 8, and 9; for since \( \frac{1}{2} = 2, \) the logarithm of 2 will be the difference of the logarithms of 10 and 5, the logarithm of 4 will be two times the logarithm of 2, the logarithm of 6 will be the sum of the logarithm of 2 and 3, and the Logarithms logarithm of 9 double the logarithm of 3. So, also having found the logarithms of 13, 17, and 19, and also of 23 and 29, they did easily get those of all the numbers between 10 and 30, by addition and subtraction only; and to having found the logarithms of other prime numbers, they got those of other numbers compounded of them.

But since the way above hinted at, for finding the logarithms of the prime numbers, is so intolerably laborious and troublesome, the more skilful mathematicians that came after the first inventors, employing their thoughts about abbreviating this method, had a vastly more easy and short way offered to them from the contemplation and mensuration of hyperbolic spaces contained between the portions of an asymptote, right lines perpendicular to it, and the curve of the hyperbola: for if ECN (Plate CLXI. fig. 6. n° 1.) be an hyperbola, and AD, AQ, the asymptotes, and AB, AP, AQ, &c. taken upon one of them, be represented by numbers, and the ordinates BC, PM, QN, &c. be drawn from the several points B, P, Q, &c. to the curve, then will the quadrilinear spaces BCMP, PMNQ, &c. viz. their numerical measures, be the logarithms of the quotients of the division of AB by AP, AP by AQ, &c. since, when AB, AP, AQ, &c. are continual proportions, the said spaces are equal, as is demonstrated by several writers concerning conic sections.

Having said that these hyperbolic spaces, numerically expressed, may be taken for logarithms, we shall next give a specimen, from the great Sir Isaac Newton, of the method how to measure these spaces, and consequently of the construction of the logarithms.

Let CA (ibid. n° 2.) = AF be = 1, and AB = Ab = x; then will \( \frac{1}{1+x} \) be BD, and \( \frac{1}{1-x} \) be bd; and putting these expressions into series, it will be \( \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5, \) &c. and \( \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + x^5, \) &c.

and \( \frac{x}{1+x} = x - x^2 + x^3 - x^4 + x^5 - x^6, \) &c. and \( \frac{x}{1-x} = x + x^2 + x^3 + x^4 + x^5 + x^6, \) &c. and taking the fluents, we shall have the area AFDB = \( \frac{x}{2} + \frac{x^3}{3} + \frac{x^5}{5}, \) &c. and the area AFdb = \( \frac{x}{2} + \frac{x^3}{3} + \frac{x^5}{5}, \) &c. and the sum bd DB = \( \frac{2x}{3} + \frac{4x^5}{5} + \frac{6x^7}{7} + \frac{8x^9}{9}, \) &c.

Now, if AB or ab be \( \frac{1}{10} - x, \) Cb being = 0.9, and CB = 1.1, by putting this value of x in the equations above, we shall have the area bd DB = 0.2006706954621511, for the terms of the series will stand as you see in this table.

| Term of the series | |-------------------| | 0.2006706954621511 |

If the parts Ad and AD of this area be added separately, logarithmically, and the lesser DA be taken from the greater dA, we shall have \( Ad - AD = x^4 + \frac{x^6}{3} + \frac{x^8}{4} + \ldots \)

= 0.0100503358535014, for the terms reduced to decimals will stand thus:

| Term of the series | |-------------------| | 0.0100000000000000 = first | | 500000000000 = second | | 333333333333 = third | | 250000000000 = fourth | | 200000000000 = fifth | | 1667 = sixth | | 14 = seventh |

Now if this difference of the areas be added to, or subtracted from their sum before found, half the aggregate, viz. 0.1053605156578263 will be the greater area Ad, and half the remainder, viz. 0.0953107198043249, will be the lesser area AD.

By the same tables, these areas AD and Ad, will be obtained also when AB = Ab are supposed to be \( \frac{1}{100} \) or CB = 1.01, and Cb = 0.99, if the numbers are but duly transferred to lower places, as

| Term of the series | |-------------------| | 0.0200000000000000 = first | | 666666666666 = second | | 400000000000 = third | | 28 = fourth |

Sum = 0.0200000667066694 = area bB.

| Term of the series | |-------------------| | 0.0001000000000000 = first | | 500000000000 = second | | 3333 = third |

Half the aggregate 0.0100503358535014 = Ad, and half the remainder, viz. 0.00995033585351681 = AD.

And so putting AB = Ab = \( \frac{1}{100} \), or CB = 1.01 and Cb = 0.99, we obtain Ad = 0.00100050003335835, and AD = 0.00099950013335835.

After the same manner, if AB = Ab, be = 0.2, or 0.02, or 0.002, these areas will arise.

\( A'd = 0.2231435513142097, \) and \( AD = 0.1823215576939546, \) or \( A'd = 0.0202027073175194, \) and \( AD = 0.1098026272961797, \) or \( A'd = 0.002002, \) and \( AD = 0.001. \)

From these areas thus found, others may be easily had from addition and subtraction only. For since \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}, \) the sum of the areas belonging to the ratios \( \frac{1}{2} \) and \( \frac{1}{2} \) (that is, insisting upon the parts of the abscissas 1, 2, 0.8; and 1, 2, 0.9) viz.

\( A'D = 0.18232, \) &c. \( A'd = 0.10536, \) &c.

Sum = 0.28768, &c.

added thus, \( 0.40546, \) &c.

Total = 0.69314, &c. = the area of AFHG, when CG is = 2. Also, since \( \frac{1}{2} \times 2 = 3, \) the sum = 1.0986122. will be the area of AFGH, when CG = 3. Again, since

\[ \frac{2x}{8} = 5, \quad \text{and} \quad 2x = 10; \quad \text{by adding} \quad Ad = 0.2231, \quad \text{etc.} \]

AD = 0.1823, etc., and AD = 0.1053, etc., together, their sum is 0.5108, etc., and this added to 1.0986, etc., the area of AFGH, when CG = 3. You will have

\[ 1.6093379124341004 = AFGH, \quad \text{when CG is } 5; \quad \text{and adding that of } 2 \text{ to this, gives } 2.302585092994057 = AFGH, \quad \text{when CG is equal to } 10; \quad \text{and since } 10 \times 10 = 100, \quad \text{and } 10 \times 100 = 1000, \quad \text{and } \sqrt{5} \times 10 \times 0.98 = 7; \quad \text{and } 10 \times 1.1 = 11, \quad \text{and } \frac{1000 \times 1.091}{7 \times 11} = 13, \quad \text{and } \frac{1000 \times 0.998}{2} = 499; \quad \text{it is plain that the area AFGH may be found by the composition of the areas found before, when CG = 100, 1000, or any other of the numbers above mentioned; and all these areas are the hyperbolic logarithms of those several numbers.}

Having thus obtained the hyperbolic logarithms of the numbers 10, 0.98, 0.99, 1.01, 1.02; if the logarithms of the four last of them be divided by the hyperbolic logarithm 2.3025850, etc., of 10, and the index 2, be added; or, which is the same thing, if it be multiplied by its reciprocal 0.434294481903 518, the value of the subtangent of the logarithmic curve, to which Briggs's logarithms are adapted, we shall have the true tabular logarithms of 98, 99, 100, 101, 102. These are to be interpolated by ten intervals, and then we shall have the logarithms of all the numbers between 980 and 1020; and all between 980 and 1000, being again interpolated by ten intervals, the table will be as it were constructed. Then from these we are to get the logarithms of all the prime numbers, and their multiples less than 100, which may be done by addition and subtraction only; for

\[ \frac{\sqrt{8} \times 1020}{9945} = 2; \quad \frac{\sqrt{8} \times 9903}{984} = 3; \quad \frac{10}{2} = 5; \quad \frac{\sqrt{98}}{2} = 7; \quad \frac{99}{9} = 11; \quad \frac{100}{7 \times 11} = 13; \quad \frac{102}{6} = 17; \quad \frac{988}{4 \times 13} = 19; \quad \frac{9936}{16 \times 27} = 23; \quad \frac{986}{2 \times 17} = 29; \quad \frac{992}{32} = 31; \quad \frac{999}{27} = 37; \quad \frac{984}{24} = 41; \quad \frac{989}{23} = 43; \quad \frac{987}{21} = 47; \quad \frac{991}{11 \times 17} = 53; \quad \frac{9971}{13 \times 13} = 59; \quad \frac{9882}{2 \times 81} = 61; \quad \frac{9949}{3 \times 49} = 67; \quad \frac{994}{14} = 71; \quad \frac{9928}{8 \times 17} = 73; \quad \frac{9954}{7 \times 18} = 79; \quad \frac{996}{12} = 83; \quad \frac{9968}{7 \times 16} = 89; \quad \frac{9894}{6 \times 17} = 97; \quad \text{and thus having the logarithms of all the numbers less than 100, you have nothing to do but interpolate the several terms, through ten intervals.}

Now the void places may be filled up by the following theorem. Let \( n \) be a number, whose logarithm is wanted; let \( x \) be the difference between that and the two nearest numbers, equally distant on each side, whose logarithms are already found; and let \( d \) be half the difference of their logarithms; then the required logarithm of the number \( n \), will be had by adding \( d + \frac{dx}{2n} + \frac{dx^3}{12n^3} \), etc., to the logarithm of the lesser number; for if the numbers are represented by \( C_p, CG, CP \), (ibid. n° 2.) and the ordinates \( f, PQ \), be raised; if \( n \) be wrote for \( CG \), and \( x \) for \( GP \), or \( Gp \), the area \( p_1QP \), or \( \frac{2x}{n} + \frac{x^3}{2n^3} \), etc., will be to the area \( p_1HG \), as the difference between the logarithms of the extreme numbers, or \( 2d \), is to the difference between the logarithms of the lesser, and of the middle one; which, therefore, will be

\[ \frac{dx}{n} + \frac{dx^3}{3n^3}, \quad \text{etc.} \]

The two first terms \( d + \frac{dx}{2n} \) of this series, being sufficient for the construction of a canon of logarithms, even to 14 places of figures, provided the number, whose logarithm is to be found, be less than 1000; which cannot be very troublesome, because \( x \) is either 1 or 2: yet it is not necessary to interpolate all the places by help of this rule, since the logarithms of numbers, which are produced by the multiplication or division of the number last found, may be obtained by the numbers whose logarithms were had before, by the addition or subtraction of their logarithms. Moreover, by the difference of their logarithms, and by their second and third differences, if necessary, the void places may be supplied more expeditiously; the rules foregoing being to be applied only where the continuation of some full places is wanted, in order to obtain these differences.

By the same method rules may be found for the intercalation of logarithms, when of three numbers the logarithm of the lesser and of the middle number are given, or of the middle number and the greater; and this although the numbers should not be in arithmetical progression. Also by pursuing the steps of this method, rules may be easily discovered for the construction of artificial lines and tangents, without the help of the natural tables. Thus far the great Newton, who says, in one of his letters to Mr Leibnitz, that he was so much delighted with the construction of logarithms, at his first setting out in those studies, that he was ashamed to tell how many places of figures he had carried them at that time: and this was before the year 1666; because, he says, the plague made him lay aside those studies, and think of other things.

Dr Keil, in his Treatise of Logarithms, at the end of his Commandine's Euclid, gives a series, by means of which may be found easily and expeditiously the logarithms of large numbers. Thus, let \( z \) be an odd number, whose logarithm is sought: then shall the numbers \( z - 1 \) and \( z + 1 \) be even, and accordingly their logarithms, and the difference of the logarithms will be had, which let be called \( y \). Therefore, also the logarithm of a number, which is a geometrical mean between \( z - 1 \) and \( z + 1 \), will be given, viz. equal to half the sum of the logarithms.

Now the series \( y \times \frac{1}{4z} + \frac{1}{24z^3} + \frac{181}{15120z^7} + \frac{25200z^9}{13} \), etc., shall be equal to the logarithm of the ratio, which the geometrical mean between the numbers \( z - 1 \) and \( z + 1 \), has to the arithmetical mean, viz. to the number \( z \). If the number exceeds 1000, the first term of the series, viz. \( \frac{1}{4z} \), is sufficient for producing the loga- Logarithms logarithm to 13 or 14 places of figures, and the second term will give the logarithm to 20 places of figures. But if \( z \) be greater than 10000, the first term will exhibit the logarithm to 18 places of figures; and so this series is of great use in filling up the chilids omitted by Mr Briggs. For example, it is required to find the logarithm of 20000; the logarithm

\[ 0.00000000542813; \quad \text{and if the logarithm of } 4.301051709302416 \text{ be added to the quotient, the sum will be} \]

\[ 4.301051709845230 = \text{the logarithm of } 20000. \]

Wherefore it is manifest, that to have the logarithm to 14 places of figures, there is no necessity of continuing out the quotient beyond 6 places of figures. But if you have a mind to have the logarithm to 10 places of figures only, the two first figures are enough. And if the logarithms of the numbers above 20000 are to be found by this way, the labour of doing them will mostly consist in setting down the numbers. This series is easily deduced from the consideration of the hyperbolic spaces aforesaid. The first figure of every logarithm towards the left hand, which is separated from the rest by a point, is called the index of that logarithm; because it points out the highest or remotest place of that number from the place of unity in the infinite scale of proportions towards the left hand; thus, if the index of the logarithm be 1, it shows that its highest place towards the left hand is the tenth place from unity; and therefore all logarithms which have 1 for their index, will be found between the tenth and hundredth place, in the order of numbers. And for the same reason, all logarithms which have 2 for their index, will be found between the hundredth and thousandth place, in the order of numbers, &c. Whence universally the index or characteristic of any logarithm is always less by one than the number of figures in whole numbers, which answer to the given logarithm; and, in decimals, the index is negative.

As all systems of logarithms whatever, are composed of similar quantities, it will be easy to form, from any system of logarithms, another system in any given ratio; and consequently to reduce one table of logarithms into another of any given form. For as any one logarithm in the given form, is to its correspondent logarithm in another form; so is any other logarithm in the given form, to its correspondent logarithm in the required form; and hence we may reduce the logarithms of lord Napier into the form of Briggs's, and contrariwise. For as 2.302585092, &c., lord Napier's logarithm of 10, is to 1.0000000000 Mr Briggs's logarithm of 10; so is any other logarithm in lord Napier's form, to the correspondent tabular logarithm in Mr Briggs's form: And because the two first numbers constantly remain the same; if lord Napier's logarithm of any one number be divided by 2.302585, &c. or multiplied by 4.342944, &c. the ratio of 1.0000, &c. to 2.30258, &c. as is found by dividing 1.00000, &c. by 2.30258, &c. the quotient in the former, and the product in the latter, will give the correspondent logarithm in Briggs's form, and the contrary. And, after the same manner, the ratio of natural logarithms to that of Briggs's will be found=868588963866.

The Use and Application of Logarithms.

It is evident, from what has been said of the con-

struction of logarithms, that addition of logarithms must be the same thing as multiplication in common arithmetic; and subtraction in logarithms the same as division; therefore, in multiplication by logarithms, add the logarithms of the multiplicand and multiplier together, their sum is the logarithm of the product.

Example. \[ \begin{array}{ccc} \text{Multiplicand} & 8.5 & 0.1294189 \\ \text{Multiplier} & 10 & 1.0000000 \\ \hline \text{Product} & 85 & 1.9294189 \\ \end{array} \]

And in division, subtract the logarithm of the divisor from the logarithm of the dividend, the remainder is the logarithm of the quotient.

Example. \[ \begin{array}{ccc} \text{Dividend} & 9712.8 & 3.9873444 \\ \text{Divisor} & 456 & 2.6589648 \\ \hline \text{Quotient} & 21.3 & 1.3283796 \\ \end{array} \]

To find the Complement of a Logarithm.

Begin at the left hand, and write down what each figure wants of 9, only what the last significant figure wants of 10; so the complement of the logarithm of 456, viz. 2.6589648, is 7.3410352.

In the rule of three. Add the logarithms of the second and third terms together, and from the sum subtract the logarithm of the first, the remainder is the logarithm of the fourth. Or, instead of subtracting a logarithm, add its complement, and the result will be the same.

To raise Powers by Logarithms.

Multiply the logarithm of the number given, by the index of the power required; the product will be the logarithm of the power sought.

Example. Let the cube of 32 be required by logarithms. The logarithm of 32=1.5051500, which multiplied by 3, is 4.5154500, the logarithm of 32768, the cube of 32. But in raising powers, viz. squaring, cubing, &c. of any decimal fraction by logarithms, it must be observed, that the first significant figure of the power be put so many places below the place of units, as the index of its logarithm wants of 10, 100, &c. multiplied by the index of the power.

To extract the Roots of Powers by Logarithms.

Divide the logarithm of the number by the index of the power, the quotient is the logarithm of the root sought.

To find mean Proportional between any two numbers.

Subtract the logarithm of the least term from the logarithm of the greatest, and divide the remainder by a number more by one than the number of means de- Logarithms desired; then add the quotient to the logarithm of the least term (or subtract it from the logarithm of the greatest) continually, and it will give the logarithms of all the mean proportionals required.

Example. Let three mean proportionals be sought, between 106 and 100.

Logarithm of 106 = 2.0253058 Logarithm of 100 = 2.0000000

Divided by 4)0.0253059(0.006326475

Logarithm of the least term 100 added 2.0000000

Logarithm of the first mean 101.4673846 2.006326475 Logarithm of the second mean 102.9563014 2.012652915 Logarithm of the third mean 104.4670483 2.018979425 Logarithm of the greatest term 106 2.0253059

The following method, communicated by Mr Thomas Atkinson, Esq. of Ballyhannon, Ireland, is much more expeditious and easy.

In any series of numbers in a geometrical progression, beginning from unity, as in the margin, the series is composed of a set of continued proportions, of which the member standing nearest to unity is the common ratio or rate of the proportion. If over or under these another series is placed, as in the example, of numbers in an arithmetical progression, beginning with nought, and whose common difference is unity, the members of this series are called indexes; for they serve to show how many successive multiplications have been made with the common rate to produce that member of the geometrical progression over which each of these indexes does severally stand.

This theory may be considered in another light. If the square root of 10 (that is, of the common rate) is found, it is a mean proportional between 1 and 10, and becomes a new common rate for a new set of continued proportions, as in the margin. And if the half of unity, which in the former case was the additional difference of the arithmetical progression, is made the additional difference of this new series, and noted as in the example, a new combination is formed of two series agreeing with the first in these remarkable properties, viz. If any two members of the geometrical progression are multiplied together, the sum of their corresponding indexes will become the index of their product; and conversely, if any one of them is divided by any other, the difference of their indexes will be found to be the index of the quotient. This theory is indefinite; and repeated extractions may be made with any proposed number of decimals, and biflection made of the corresponding indexes, until one has no more number to work with; and each of the mean proportionals thus found between 1 and 10, will be found a member of every new geometrical progression formed by every smaller root; and consequently all the roots thus found, together with their corresponding indexes, have,

.0000087837, .0000038147 its logm .0000025175, its logm

Thus knowing that 0.0000025175, or such like, is the logm. of the last quotient, one may have that of 2, if he will but call to mind the following circumstances.

In every case of division, if he has logarithms of quotient and divisor, he has also that of the dividend, by adding the two first together; if he has the logarithm of the dividend, and that of either the divisor or Logarithms or quotient; he may find that of the other; so he has only to subtract what he knows from the logarithm of the dividend, the remainder is what he wants, and lastly, that in every division he made, he took one number from the table of roots whose logarithm is known, being noted in the table, and which he made use of as his direction either as a dividend or a divisor: From these circumstances, one may, by the help of the logarithm just found, discover the logarithm of that number of the last division, whether it be dividend or divisor, which was the quotient of the preceding division; and thus, tracing his own work backwards by his notes from quotient to quotient, he they ever so few or ever so many, he will come at last by addition and subtraction to the logarithm of the proposed number.

By this method, the logarithm of any number within the compass of the table of roots may be found; if a greater is proposed, suppose 9495, it must be made 9495, and its logarithm found; then it must be restored to the proposed form, and have a proper index noted before the decimals just found. How to do this is too well known to have occasion to mention it here.

\[-3.301029995664 \text{ the logarithm of the fraction given.}\]

\[7 \text{ the power to which it is to be raised.}\]

\[-19.107209969648 \text{ the logarithm of the answer.}\]

This differs from the like work in whole numbers only in this, that, in multiplying the decimals, one has at last 2 to be carried from them to the whole numbers; this is to be considered as +2, then \(3 \times 7 = -21\), and \(-21 + 2 = -19\) to be noted the index of the answer. Extraction of the roots is only the converse of this. Suppose \(-19.107209969648\) given, to find that root whose exponent number is 7. As 7 is the exponent number here, one may in his mind multiply it by 2 for a trial, as in common division; but the product = 14 being less than 19, must be rejected; then he may try it with 3; this yields 21 for a product. This 3 must be noted with a negative sign for the index of the new logarithm. Then, on comparing 19 with 21, the difference is 2. This 2 must be carried as 20 to the decimals, and one must from that carry on the division of the decimals with 7 for a divisor, as is usually done in other cases.

Another Example.

Suppose \(-1.4771212545\) given, to extract the root of its 5th power.

\[-1.8954252109 \text{ the logarithm of the root.}\]

For 5, the exponent of the root \(x^5\) is greater than the index of the given logarithm, and 4 is the remainder. Then \(-1\) becomes the index of the logarithm of the root; and 4 = the overplus, is to be carried as 40 to the decimals; and from that, division is to be made with 5 as a divisor for the rest of the work.

LOGIC LOGIC,

The art of thinking and reasoning justly; or, it may be defined the science or history of the human mind, inasmuch as it traces the progress of our knowledge from our first and most simple conceptions through all their different combinations, and all those numerous deductions that result from variously comparing them one with another.

The precise business of logic, therefore, is to explain the nature of the human mind, and the proper manner of conducting its several powers, in order to the attainment of truth and knowledge. It lays open those errors and mistakes we are apt, through inattention, to run into; and teaches us how to distinguish between truth, and what only carries the appearance of it. By this means we grow acquainted with the nature and force of the understanding; see what things lie within its reach; where we may attain certainty and demonstration; and when we must be contented with probability.

This science is generally divided into four parts, viz. Perception, Judgment, Reasoning, and Method. This division comprehends the whole history of the sensations and operations of the human mind.

PART I.

OF PERCEPTION.

We find ourselves surrounded with a variety of objects, which acting differently upon our senses, convey distinct impressions into the mind, and thereby rouse the attention and notice of the understanding. By reflecting too on what passes within us, we become sensible of the operations of our own minds, and attend to them as a new set of impressions. But in all this there is only bare consciousness. The mind, without proceeding any farther, takes notice of the impressions that are made upon it, and views things in order, as they present themselves one after another. This attention of the understanding to the object acting upon it, whereby it becomes sensible of the impressions they make, is called by logicians perception; and the notices themselves, as they exist in the mind, and are there treasured up to be the materials of thinking and knowledge, are distinguished by the name of ideas. Having flown at large, in the article Metaphysics, how the mind being furnished with ideas, contrives to diversify and enlarge its stock; we have here chiefly to consider the means of making known our thoughts to others; that we may not only understand how knowledge is acquired, but also in what manner it may be communicated with the greatest certainty and advantage.

CHAP. I. Of Words, considered as the Signs of our Ideas.

Our ideas, though manifold and various, are nevertheless all within our own breasts, invisible to others, nor can of themselves be made appear. But God, designing us for society, and to have fellowship with those of our kind, has provided us with organs fitted to frame articulate sounds, and given us also a capacity of using those sounds as signs of internal conceptions. Hence spring words and language; for, having once pitched upon any sound to stand as the mark of an idea in the mind, custom by degrees establishes such a connection between them, that the appearance of the idea in the understanding always brings to our remembrance the sound or name by which it is expressed; as in like manner the hearing of the sound never fails to excite the idea for which it is made to stand. And thus it is easy to conceive how a man may record his own thoughts, and bring them again into view in any succeeding period of life. For this connection being once settled, as the same sounds will always serve to excite the same ideas; if he can but contrive to register his words in the order and disposition in which the present train of his thoughts present them to his imagination, it is evident he will be able to recall these thoughts at pleasure, and that too in the very manner of their first appearance. Accordingly we find, that the inventions of writing and printing, by enabling us to fix and perpetuate such perishable things as sounds, have also furnished us with the means of giving a kind of permanency to the transactions of the mind, insomuch that they may be in the same manner subjected to our review, as any other abiding objects of nature.

II. But, besides the ability of recording our own thoughts, there is this farther advantage in the use of external signs, that they enable us to communicate our sentiments to each other, and also receive information of what passes in their breasts. For any number of men, having agreed to establish the same sounds as signs of the same ideas, it is apparent that the repetition of these sounds must excite the like perceptions in each, and create a perfect correspondence of thoughts. When, for instance, any train of ideas succeed one another in my mind, if the names by which I am wont to express them have been annexed by those with whom I converse to the very same set of ideas, nothing is more evident, than that, by repeating those names according to the tenor of my present conceptions, I shall raise in their minds the same course of thought as has taken possession of my own. Hence, by barely attending to what passes within themselves, they will also become acquainted with the ideas in my understanding, and have them in a manner laid before their view. So that we here clearly perceive how a man may communicate his sentiments, knowledge, and discoveries to others, if the language in which he converses be extensive enough to mark all the ideas and transactions of his mind. But as this is not always the case, and men are often obliged to invent vent terms of their own, to express new views and conceptions of things; it may be asked, how in these circumstances we can become acquainted with the thoughts of another, when we make use of words, to which we have never annexed any ideas, and that of course can raise no perceptions in our minds. In order to unveil this mystery, and give some little insight into the foundation, growth, and improvement of language, the following observations will be found of considerable moment.

III. First, That no word can be to any man the sign of an idea, till that idea comes to have a real existence in his mind. For names, being only so far intelligible as they denote known internal conceptions, where they have none such to answer them, there they are plainly founds without signification, and of course convey no instruction or knowledge. But no sooner are the ideas to which they belong raised in the understanding, than, finding it easy to connect them with the established names, we can join in any agreement of this kind made by others, and thereby enjoy the benefit of their discoveries. The first thing therefore to be considered is, how these ideas may be conveyed into the mind; that being there, we may learn to connect them with their appropriated founds, and so become capable of understanding others when they make use of these founds in laying open and communicating their thoughts. Now, to comprehend this distinctly, it will be necessary to attend to the division of our ideas into simple and complex, (see Metaphysics). And first, as for our simple ideas; they can find no admission into the mind, but by the two original fountains of knowledge, sensation and reflection. If therefore any of these have as yet no being in the understanding, it is impossible by words or a description to excite them there. A man who had never felt the impression of heat, could not be brought to comprehend that sensation by anything we might say to explain it. If we would really produce the idea in him, it must be by applying the proper object to his senses, and bringing him within the influence of a hot body. When this is done, and experience has taught him the perception to which men have annexed the name heat, it then becomes to him the sign of that idea, and he thenceforth understands the meaning of the term, which, before, all the words in this world would not have been sufficient to convey into his mind. The case is the same in respect of light and colours. A man born blind, and thereby deprived of the only conveyance for the ideas of this class, can never be brought to understand the names by which they are expressed. The reason is plain: they stand for ideas that have no existence in his mind; and as the organ appropriated to their reception is wanting, all other contrivances are vain, nor can they by any force of description be raised in his imagination. But it is quite otherwise in our complex notions. For these being no more than certain combinations of simple ideas, put together in various forms; if the original ideas out of which the collections are made have already got admission into the understanding, and the names serving to express them are known; it will be easy, by enumerating the several ideas concerned in the composition, and marking the order and manner in which they are united, to raise any complex conception in the mind. Thus the idea answering to the word rainbow may be readily excited in the imagination of another who has never seen the appearance itself, by barely describing the figure, largeness, position, and order of colours; if we suppose these several simple ideas, with their names, sufficiently known to him.

IV. And this leads to a second observation upon this subject, namely, That words standing for complex ideas are all definable, but those by which we denote simple ideas are not; for the perceptions of this latter class, having no other entrance into the mind than by sensation or reflection, can only be got by experience, from the several objects of nature, proper to produce those perceptions in us. Words indeed may very well serve to remind us of them, if they have already found admission into the understanding, and their connection with the established names is known; but they can never give them their original being and existence there. And hence it is, that when any one asks the meaning of a word denoting a simple idea, we pretend not to explain it to him by a definition, well knowing that to be impossible; but supposing him already acquainted with the idea, and only ignorant of the name by which it is called, we either mention it to him by some other name, with which we presume he knows its connection, or appeal to the object where the idea itself is found. Thus was anyone to ask the meaning of the word white, we should tell him it stood for the same idea as albus in Latin, or blanc in French; or, if we thought him a stranger to these languages, might appeal to an object producing the idea, by saying it denoted the colour we observe in sugar or milk. But this is by no means a definition of the word, exciting a new idea in his understanding; but merely a contrivance to remind him of a known idea, and teach him its connection with the established name. For if the ideas after which he enquires have never yet been raised in his mind; as suppose one who had seen no other colours than black and white, should ask the meaning of the word scarlet; it is easy to perceive, that it would be no more possible to make him comprehend it by words, or a definition, than to discourse the same perception into the imagination of a man born blind. The only method in this case is, to present some object, by looking at which the perception itself may be excited, and thus he will learn both the name and the idea together.

V. But how comes it to pass that men agree in the names of their simple ideas, feeling they cannot view the perceptions in one another's minds, nor make known these perceptions by words to others? The effect is produced by experience and observation. Thus finding, for instance, that the name of heat is annexed to that impression which men feel when they approach the fire, I make it also the sign of the idea excited in me by such an approach, nor have any doubt but it denotes the same perception in my mind as in theirs. For we are naturally led to imagine, that the same objects operate alike upon the organs of the human body, and produce an uniformity of sensations. No man fancies, that the idea raised in him by the taste of sugar, and which he calls sweetness, differs from that excited in another by the like means; or that wormwood, to whose relish he has given the epithet bitter, produces in another the sensation which he denotes by the word fustet. Presuming therefore upon this conformity of perceptions, when they arise from the same objects, we easily agree as to the names of our simple ideas; and if at any time, by a more narrow scrutiny into things, new ideas of this class come in our way, which we choose to express by terms of our own invention; these names are explained, not by a definition, but by referring to the objects whence the ideas themselves may be obtained.

VI. Being in this manner furnished with simple ideas, and the names by which they are expressed, the meaning of terms that stand for complex ideas is easily got; because the ideas themselves answering to these terms may be conveyed into the mind by definitions. For our complex notions are only certain combinations of simple ideas. When therefore these are enumerated, and the manner in which they are united into one conception, explained, nothing more is wanting to raise that conception in the understanding; and thus the term denoting it comes of course to be understood. And here it is worth while to reflect a little upon the wise contrivance of nature, in thus furnishing us with the very aptest means of communicating our thoughts. For were it not so ordered, that we could thus convey our complex ideas from one to another by definitions, it would in many cases be impossible to make them known at all. This is apparent in those ideas which are the proper work of the mind. For as they exist only in the understanding, and have no real objects in nature in conformity to which they are framed; if we could not make them known by description, they must lie for ever hid within our own breasts, and be confined to the narrow acquaintance of a single mind. All the fine scenes that arise from time to time in the poet's fancy, and by his lively painting give such entertainment to his readers; were he destitute of this faculty of laying them open to the view of others by words and description, could not extend their influence beyond their own imagination, or give joy to any but the original inventor.

VII. There is this farther advantage, in the ability we enjoy of communicating our complex notions by definitions; that as these make by far the largest class of our ideas, and most frequently occur in the progress and improvement of knowledge, so they are by these means imparted with the greatest readiness, than which nothing could tend more to the increase and spreading of science: for a definition is soon perused; and if the terms of it are well understood, the idea itself finds an easy admission into the mind. Whereas in simple perceptions, where we are referred to the objects producing them, if these cannot be come at; as is sometimes the case, the names by which they are expressed must remain empty sounds. But new ideas of this class occurring very rarely in the sciences, they seldom create any great obstruction. It is otherwise with our complex notions; for every step we take leading us into new combinations and views of things, it becomes necessary to explain these to others, before they can be made acquainted with our discoveries, and as the manner of definitions is easy, requiring no apparatus but that of words, which are always ready, and at hand; hence we can with the least difficulty remove such obstacles as might arise from terms of our own invention, when they are made to stand for new, complex ideas suggested to the mind by some present train of thinking. And thus at last we are let into the mystery hinted at in the beginning of this chapter, viz. how we may become acquainted with the thoughts of another, when he makes use of words to which we have as yet joined no ideas. The answer is obvious from what has been already said. If the terms denote simple perceptions, he must refer us to these objects of nature whence the perceptions themselves are to be obtained; but, if they stand for complex ideas, their meaning may be explained by a definition.

CHAP. II. Of Definitions.

I. A definition is the unfolding of some conception of the mind, answering to the word or term made use of as the defined sign of it. Now as, in exhibiting any idea to another, it is necessary that the description be such as may excite that precise idea in his mind; hence it is plain that definitions, properly speaking, are not arbitrary, but confined to the representing of certain determinate settled notions, such namely as are annexed by the speaker or writer to the words he uses. As nevertheless it is universally allowed that the signification of words is perfectly voluntary, and not the effect of any natural and necessary connection between them and the ideas for which they stand; some may perhaps wonder why definitions are not so too. In order therefore to unravel this difficulty, and shew distinctly what is and what is not arbitrary in speech, we must carefully distinguish between the connection of our words and ideas, and the unfolding of the ideas themselves.

II. First, as to the connection of our words and ideas; this, it is plain, is a purely arbitrary institution. When, for instance, we have in our minds the idea of any particular species of metals, the calling it by the name gold is an effect of the voluntary choice of men speaking the same language, and not of any peculiar aptness in that found to express that idea. Other nations we find make use of different sounds, and with the same effect. Thus aurum denotes that idea in Latin, and or in French; and even the word gold itself would have as well served to express the idea of that metal which we call silver, had custom in the beginning established it.

III. But although we are thus entirely at liberty in connecting any idea with any sound, yet it is quite otherwise in unfolding the ideas themselves. For every idea having a precise appearance of its own, by which it is distinguished from every other idea; it is manifest, that in laying it open to others, we must study such a description as shall exhibit that peculiar appearance. When we have formed to ourselves the idea of a figure bounded by four equal sides, joined together at right angles, we are at liberty to express that idea by any sound, and call it either a square or a triangle. But which ever of these names we use, so long as the idea is the same, the description by which we would signify it to another must be so too. Let it be called square or triangle, it is still a figure having four four equal sides, and all its angles right ones. Hence we clearly see what is and what is not arbitrary in the use of words. The establishing any sound as the mark of some determinate idea in the mind, is the effect of free choice, and a voluntary combination among men. And as different nations make use of different sounds to denote the same ideas, hence proceeds all that variety of languages which we meet with in the world. But when a connection between our ideas and words is once settled, the unfolding of the idea answering to any word, which properly constitutes a definition, is by no means an arbitrary thing. For here we are bound to exhibit that precise conception which either the use of language, or our own particular choice, hath annexed to the term we use.

IV. And thus it appears, that definitions, considered as descriptions of ideas in the mind, are steady and invariable, being bounded to the representation of these precise ideas. But then, in the application of definitions to particular names, we are altogether left to our own free choice. Because as the connecting of any idea with any sound is a perfectly arbitrary institution; the applying the description of that idea to that sound must be so too. When therefore logicians tell us that the definition of the name is arbitrary, they mean no more than this; that as different ideas may be connected with any term, according to the good pleasure of him that uses it, in like manner may different descriptions be applied to the term suitable to the ideas so connected. But this connection being settled, and the term considered as the sign of some fixed idea in the understanding, we are no longer left to arbitrary explications, but must study such a description as corresponds with that precise idea. Now this alone, according to what has been before laid down, ought to be accounted a definition. What seems to have occasioned no small confusion in this matter, is, that many explanations of words where no idea is unfolded, but merely the connection between some word and idea asserted, have yet been dignified with the name of definitions. Thus, when we say that a clock is an instrument by which we measure time; that is by some called a definition; and yet it is plain that we are beforehand supposed to have an idea of this instrument, and only taught that the word clock serves in common language to denote that idea. By this rule all explications of words in our dictionaries will be definitions, nay, the names of even simple ideas may be thus defined. White, we may say, is the colour we observe in snow or milk; hear the sensation produced by approaching the fire; and so in innumerable other instances. But these, and all others of the like kind, are by no means definitions, exciting new ideas in the understanding, but merely contrivances to remind us of known ideas, and teach their connection with the established names.

V. But now in definitions properly so called, we first consider the term we use, as the sign of some inward conception, either annexed to it by custom, or our own free choice; and then the business of the definition is to unfold and explicate that idea. As therefore the whole art lies in giving just and true copies of our ideas; a definition is then said to be made perfect, when it serves distinctly to excite the idea described in the mind of another, even supposing him before wholly unacquainted with it. This point settled, let us next inquire what those ideas are which are capable of being thus unfolded? And in the first place it is evident, that all our simple ideas are necessarily included. We have seen already that experience alone is to be consulted here, inasmuch that if either the objects whence they are derived come not in our way, or the avenues appointed by nature for their reception are wanting, no description is sufficient to convey them into the mind. But where the understanding is already supplied with these original and primitive conceptions, as they may be united together in an infinity of different forms; so may all their several combinations be distinctly laid open, by enumerating the simple ideas concerned in the various selections, and tracing the order and manner in which they are linked one to another. Now these combinations of simple notices constitute what we call our complex notions; whence it is evident that complex ideas, and those alone, admit of that kind of description which goes by the name of a definition.

VI. Definitions, then, are pictures or representations of our ideas; and as these representations are then only possible when the ideas themselves are complex, it is obvious to remark, that definitions cannot have place but where we make use of terms standing for such complex ideas. But our complex ideas, being as we have said nothing more than different combinations of simple ideas; we then know and comprehend them perfectly, when we know the several simple ideas of which they consist, and can put them together in our minds as is necessary towards the framing of that peculiar connection which gives every idea its distinct and proper appearance.

VII. Two things are therefore required in every definition. First, that all the original ideas, out of which the complex one is formed, be distinctly enumerated. Secondly, that the order and manner of combining them into one conception be clearly explained. Where a definition has these requisites, nothing is wanting to its perfection; because every one who reads it and understands the terms, seeing at once what ideas he is to join together, and also in what manner, can at pleasure form in his own mind the complex conception answering to the term defined. Let us, for instance, suppose the word square to stand for that idea by which we represent to ourselves a figure whose sides subtend quadrants of a circumscribed circle. The parts of this idea are the sides bounding the figure. These must be four in number, and all equal among themselves, because they are each to subtend a fourth part of the same circle. But, besides these component parts, we must also take notice of the manner of putting them together, if we would exhibit the precise idea for which the word square here stands. For four equal right lines, any-how joined, will not subtend quadrants of a circumscribed circle. A figure with this property must have its sides standing also at right angles. Taking in therefore this last consideration reflecting the manner of combining the parts, the idea is fully described, and the definition thereby rendered complete. For a figure bounded by four equal sides, joined together at right angles, has VIII. It will now be obvious to every one in what manner we ought to proceed, in order to arrive at just and adequate definitions. First, we are to take an exact view of the idea to be described, trace it to its original principles, and mark the several simple perceptions that enter into the composition of it. Secondly, we are to consider the particular manner in which these elementary ideas are combined, in order to the forming of that precise conception for which the term we make use of stands. When this is done, and the idea wholly unravelled, we have nothing more to do than fairly transcribe the appearance it makes to our own minds. Such a description, by distinctly exhibiting the order and number of our primitive conceptions, cannot fail to excite at the same time in the mind of every one that reads it, the complex idea resulting from them; and therefore attains the true and proper end of a definition.

CHAP. III. Of the Composition and Resolutions of our Ideas, and the Rules of Definition thence arising.

I. The rule laid down in the foregoing chapter is general, extending to all possible cases; and is indeed that to which alone we can have recourse, where any doubt or difficulty arises. It is not however necessary that we should practise it in every particular instance. Many of our ideas are extremely complicated, in such a manner that to enumerate all the simple perceptions out of which they are formed, would be a very troublesome and tedious work. For this reason logicians have established certain compendious rules of defining, of which it may not be amiss here to give some account. But in order to the better understanding of what follows, it will be necessary to observe that there is a certain gradation in the composition of our ideas. The mind of man is very limited in its views, and cannot take in a great number of objects at once. We are therefore fain to proceed by steps, and make our first advances subservient to those which follow. Thus in forming our complex notions, we begin at first with but a few simple ideas, such as we can manage with ease, and unite them together into one conception. When we are provided with a sufficient stock of these, and have by habit and use rendered them familiar to our minds, they become the component parts of other ideas still more complicated, and form what we may call a second order of compound notions. This process, as is evident, may be continued to any degree of composition we please, mounting from one stage to another, and enlarging the number of combinations.

II. But now in so far as this kind, whoever would hence ideas of this class acquaint his mind, forming therewith the last and highest order of ideas, finds it much the most expedient method to proceed gradually through all the intermediate steps. For, was he to take any very compound idea to pieces, and, without regard to the several classes of simple perceptions that have already been formed into distinct combinations, break it at once into its original principles, the number would be so great as perfectly to confound the imagination, and overcome the utmost reach and capacity of the mind. When we see a prodigious multitude of men jumbled together in crowds, without order or any regular position, we find it impossible to arrive at an exact knowledge of their number. But if they are formed into separate battalions, and so stationed as to fall within the leisure survey of the eye; by viewing them successively and in order, we come to an easy and certain determination. It is the same in our complex ideas. When the original perceptions, out of which they are framed, are very numerous, it is not enough that we take a view of them in loose and scattered bodies; we must form them into distinct classes, and unite these classes in a just and orderly manner, before we can arrive at a true knowledge of the compound notions resulting from them.

III. This gradual progress of the mind to its compound notions, through a variety of intermediate steps, plainly points out the manner of conducting the definitions, by which these notions are conveyed into the minds of others. For as the series begins with simple and easy combinations, and advances through a succession of different orders, rising one above another in the degree of composition; it is evident that, in a train of definitions expressing these ideas, a like gradation is to be observed. Thus the complex ideas of the lowest order can no otherwise be described than by enumerating the simple ideas out of which they are made, and explaining the manner of their union. But then in the second, or any other succeeding order; as they are formed out of those gradual combinations, and constitute the inferior classes, it is not necessary, in describing them, to mention one by one all the simple ideas of which they consist. They may be more distinctly and briefly unfolded, by enumerating the compound ideas of a lower order, from whose union they result, and which are all supposed to be already known in consequence of previous definitions. Here then it is that the logical method of defining takes place; which that it may be the better understood, we shall explain somewhat more particularly the several steps and gradations of the mind in compounding its ideas, and thence deduce that peculiar form of a definition which logicians have thought fit to establish.

IV. All the ideas we receive from the several objects of nature that surround us, represent distinct individuals. These individuals, when compared together, are found in certain particulars to resemble. Hence, by collecting the resembling particulars into one conception, we form the notion of a species. And here let it be observed, that this last idea is less complicated than that by which we represent any of the particular objects contained under it. For the idea of the species excludes the peculiarities of the several individuals, and retains only such properties as are common to them all. Again, by comparing several species together, and observing their resemblance, we form the idea of a genus; where, in the same manner as before, the composition is lessened, because we leave out what is peculiar to the several species compared, and retain only the particulars wherein they agree. It is easy to conceive the mind proceeding thus from one step to another, and advancing through its several classes of general notions, until at last it comes comes to the highest genus of all, denoted by the word being, where the bare idea of existence is only concerned.

V. In this procedure we see the mind unravelling a complex idea, and tracing it in the ascending scale, from greater or less degrees of composition, until it terminates in one simple perception. If now we take the series the contrary way, and, beginning with the last or highest genus, carry our view downwards, through all the inferior genera and species, quite to the individuals, we shall thereby arrive at a distinct apprehension of the conduct of the understanding in compounding its ideas. For, in the several classes of our perceptions, the highest in the scale is for the most part made up of but a few simple ideas, such as the mind can take in and survey with ease. This first general notion, when branched out into the different subdivisions contained under it, has in every one of them something peculiar, by which they are distinguished among themselves; infomuch that, in descending from the genus to the species, we always superadd some new idea, and thereby increase the degree of composition. Thus the idea denoted by the word figure is of a very general nature, and composed of but few simple perceptions, as implying no more than space everywhere bounded. But if we descend farther, and consider the boundaries of this space, as that they may be either lines or surface, we fall into the several species of figure. For where the space is bounded by one or more surfaces, we give it the name of a solid figure; but where the boundaries are lines, it is called a plain figure.

VI. In this view of things it is evident, that the species is formed by superadding a new idea to the genus. Here, for instance, the genus is circumscribed space. If now to this we superadd the idea of a circumference by lines, we frame the notion of that species of figures which are called plain; but if we conceive the circumference to be by surfaces, we have the species of solid figures. This superadded idea is called the specific difference, not only as it serves to divide the species from the genus, but because, being different in all the several subdivisions, we thereby also distinguish the species one from another. And as it is likewise that conception, which, by being joined to the general idea, compleats the notion of the species; hence it is plain, that the genus and specific difference are to be considered as the proper and constituent parts of the species. If we trace the progress of the mind still farther, and observe it advancing through the inferior species, we shall find its manner of proceeding to be always the same. For every lower species is formed by superadding some new idea to the species next above it; infomuch that in this descending scale of our perceptions, the understanding passes through different orders of complex notions, which become more and more complicated at every step it takes. Let us resume here, for instance, the species of plain figures. They imply no more than space bounded by lines. But if we take in an additional consideration of the nature of these lines, as whether they are right or curved, we fall into the subdivisions of plain figure, distinguished by the names of rectilinear, curvilinear, and mixtilinear.

VII. And here we are to observe, that though plain figures, when considered as one of those branches that come under the notion of figure in general, take the name of a species; yet compared with the classes species, by superadding the specific difference, they really become a genus, of which the before mentioned subdivisions constitute the several species. These species, in the same manner as in the case of plain and solid figures, consist of the genus and specific difference as their constituent parts. For in the curvilinear kind, the curvity of the lines bounding the figure makes what is called the specific difference; to which if we join the genus, which here is a plain figure, or space circumscribed by lines, we have all that is necessary towards compleating the notion of this species. We are only to take notice, that this last subdivision, having two genera above it, viz., plain figure, and figure in general; the genus joined with the specific difference, in order to constitute the species of curvilinears, is that which lies nearest to the said species. It is the notion of plain figure, and not of figure in general, that joined with the idea of curvity makes up the complex conception of curve-lined figures. For in this descending scale of our ideas, figure in general, plain figures, curve-lined figures, the two first are considered as genera in respect of the third; and the second in order, or that which stands next to the third, is called the nearest genus. But now as it is this second idea, which, joined with the notion of curvity, forms the species of curve-lined figures; it is plain, that the third or last idea in the series is made up of the nearest genus and specific difference. This rule holds invariably, however far the series is continued; because, in a train of ideas thus succeeding one another, all that precede the last are considered as so many genera in respect of that last; and the last itself is always formed by superadding the specific difference to the genus next it.

VIII. Here then we have an universal description, applicable to all our ideas of whatever kind, from the highest genus to the lowest species. For, taking them in order downwards from the said general idea, they every where consist of the genus proximum, and differentia specifica, as logicians love to express themselves. But when we come to the lowest species of all, comprehending under it only individuals, the superadded idea, by which these individuals are distinguished one from another, no longer takes the name of the specific difference. For here it serves not to denote distinct species, but merely a variety of individuals, each of which, having a particular existence of its own, is therefore numerically different from every other of the same kind. We must observe that in this last case, logicians choose to denote the superadded idea by the name of the numerical difference; infomuch that, as the idea of a species is made up of the nearest genus and specific difference, so the idea of an individual consists of the lowest species and numeric difference. Thus the circle is a species of curve-lined figures, and what we call the lowest species, as comprehending under it only individuals. Circles in particular are distinguished from one another by the length and position of their diameters. The length therefore and position of the diameter of a circle is what logicians call the numerical difference; because, these being given, the circle itself may be described, and an individual thereby constituted.

IX. Thus the mind, in compounding its ideas, begins, we see, with the most general notions, which, confining of but a few simple notices, are easily combined and brought together into one conception. Thence it proceeds to the species comprehended under this general idea, and these are formed by joining together the genus and specific difference. And as it often happens, that these species may be still farther subdivided, and run on in a long series of continued gradations, producing various orders of compound perceptions; so all these several orders are regularly and successively formed by annexing in every step the specific difference to the nearest genus. When by this method of procedure we are come to the lowest order of all, by joining the species and numeric difference we frame the ideas of individuals. And here the series necessarily terminates, because it is impossible any farther to bound or limit our conceptions. This view of the composition of our ideas, representing their constituent parts in every step of the progression, naturally points out the true and genuine form of a definition. For as definitions are no more than descriptions of the ideas for which the terms defined stand; and as ideas are then described, when we enumerate distinctly and in order the parts of which they consist; it is plain that, by making our definitions follow one another according to the natural train of our conceptions, they will be subject to the same rules, and keep pace with the ideas they describe.

X. As therefore the first order of our compound notions, or the ideas that constitute the highest genera in the different scales of perception, are formed by uniting together a certain number of simple notices; so the terms expressing these genera are defined by enumerating the simple notices so combined. And as the species comprehended under any genus, or the complex ideas of the second order, arise from superadding the specific difference to the said general idea; so the definition of the names of the species is abolished, in a detail of the ideas of the specific difference, connected with the term of the genus. For the genus having been before defined, the term by which it is expressed stands for a known idea, and may therefore be introduced into all subsequent definitions, in the same manner as the names of simple perceptions. It will now be sufficiently obvious, that the definitions of all the succeeding orders of compound notions will everywhere consist of the term of the nearest genus, joined with an enumeration of the ideas that constitute the specific difference; and that the definition of individuals unites the names of the lowest species with the terms by which we express the ideas of the numeric difference.

XI. Here then we have the true and proper form of a definition, in all the various orders of conception. This is that method of defining which is commonly called logical, and which we see is perfect in its kind, inasmuch as it presents a full and adequate description of the idea for which the term defined stands.

PART II.

OF JUDGMENT.

CHAP. I. Of the Grounds of human Judgment.

The mind being furnished with ideas, its next step in the way to knowledge is, the comparing these ideas together, in order to judge of their agreement or disagreement. In this joint view of our ideas, if the relation is such as to be immediately discoverable by the bare inspection of the mind, the judgments thence obtained are called intuitive; from a word that denotes to look at; for in this case, a mere attention to the ideas compared suffices to let us see how far they are connected or disjoined. Thus, that the Whole is greater than any of its Parts, is an intuitive judgment, nothing more being required to convince us of its truth, than an attention to the ideas of whole and part. And this too is the reason why we call the act of the mind, forming these judgments, intuition; as it is indeed no more than an immediate perception of the agreement or disagreement of any two ideas.

II. But here it is to be observed, that our knowledge of this kind respects only our ideas, and the relations between them; and therefore can serve only as a foundation to such reasonings as are employed in investigating these relations. Now it so happens, that many of our judgments are conversant about facts, and the real existence of things, which cannot be traced by the bare contemplation of our ideas. It does not follow, because I have the idea of a circle in my mind, that therefore a figure answering to that idea has a real existence in nature. I can form to myself the notion of a centaur, or golden mountain, but never imagine on that account, that either of them exist. What then are the grounds of our judgment in relation to facts? experience and testimony. By experience we are informed of the existence of the several objects which surround us, and operate upon our senses. Testimony is of a wider extent, and reaches not only to objects beyond the present sphere of our observation, but also to facts and transactions, which being now past, and having no longer any existence, could not without this conveyance have fallen under our cognizance.

III. Here we have three foundations of human judgment, from which the whole system of our knowledge may with ease and advantage be derived. First, intuition, which respects our ideas themselves, and their relations, and is the foundation of that species of reasoning which we call demonstration. For whatever is deduced from our intuitive perceptions, by a clear and connected series of proofs, is said to be demonstrated, and produces absolute certainty in the mind. Hence the knowledge obtained in this manner is what we properly term science; because in ev- ry step of the procedure it carries its own evidence along with it, and leaves no room for doubt or hesitation. And what is highly worthy of notice; as the truths of this class express the relation between our ideas, and the same relations must ever and invariably subsist between the same ideas, our deductions in the way of science constitute what we call eternal, necessary, and immutable truths. If it be true that the whole is equal to all its parts, it must be so unchangeably; because the relation of equality being attached to the ideas themselves, must ever intervene where the same ideas are compared. Of this nature are all the truths of natural religion, morality, and mathematics, and, in general, whatever may be gathered from the bare view and consideration of our ideas.

IV. The second ground of human judgment is experience; from which we infer the existence of those objects that surround us, and fall under the immediate notice of our senses. When we see the sun, or cast our eyes towards a building, we not only have ideas of these objects within ourselves, but ascribe to them a real existence out of the mind. It is also by the information of the senses, that we judge of the qualities of bodies; as when we say that snow is white, fire hot, or steel hard. For as we are wholly unacquainted with the internal structure and constitution of the bodies that produce these sensations in us, nay, and are unable to trace any connection between that structure and the sensations themselves, it is evident, that we build our judgments altogether upon observation, attributing to bodies such qualities as are answerable to the perceptions they excite in us. But this is not the only advantage derived from experience, for to that too are we indebted for all our knowledge regarding the co-existence of sensible qualities in objects, and the operations of bodies one upon another. Ivory, for instance, is hard and elastic; this we know by experience, and indeed by that alone. For, being altogether strangers to the true nature both of elasticity and hardness, we cannot by the bare contemplation of our ideas determine how far the one necessarily implies the other, or whether there may not be a repugnance between them. But when we observe them to exist both in the same object, we are then assured from experience, that they are not incompatible; and when we also find, that a stone is hard and not elastic, and that air though elastic is not hard, we also conclude upon the same foundation, that the ideas are not necessarily conjoined, but may exist separately in different objects. In like manner with regard to the operations of bodies one upon another, it is evident, that our knowledge this way is all derived from observation. Aqua regia dissolves gold, as has been found by frequent trial, nor is there any other way of arriving at the discovery. Naturalists may tell us, if they please, that the parts of aqua regia are of a texture apt to infuse between the corpuscles of gold, and thereby loosen and shake them asunder. If this is a true account of the matter, it will notwithstanding be allowed, that our conjecture in regard to the conformation of these bodies is deduced from the experiment, and not the experiment from the conjecture. It was not from any previous knowledge of the intimate structure of aqua regia and gold, and the aptness of their parts to act or to be acted upon, that we came by the conclusion above-mentioned. The internal constitution of bodies is in a manner wholly unknown to us; and could we even surmount this difficulty, yet as the separation of the parts of gold implies something like an active force in the menstruum, and we are unable to conceive how it comes to be possessed of this activity; the effect must be owned to be altogether beyond our comprehension. But when repeated trials had once confirmed it, inasmuch that it was admitted as an established truth in natural knowledge, it was then easy for men to spin out theories of their own invention, and contrive such a structure of parts, both for gold and aqua regia, as would best serve to explain the phenomenon upon the principles of that system of philosophy they had adopted.

V. From what has been said it is evident, that as intuition is the foundation of what we call scientific knowledge, so is experience of natural. For this last being wholly taken up with objects of sense, or those bodies that constitute the natural world; and their properties, as far as we can discover them, being to be traced only by a long and painful series of observations; it is apparent, that in order to improve this branch of knowledge, we must betake ourselves to the method of trial and experiment.

VI. But though experience is what we may term the immediate foundation of natural knowledge, yet with respect to particular persons its influence is very narrow and confined. The bodies that surround us are numerous, many of them lie at a great distance, and some quite beyond our reach. Life is too short, and too crowded with cares, that but little time is left for any single man to employ himself in unfolding the mysteries of nature. Hence it is necessary to admit many things upon the testimony of others, which by this means becomes the foundation of a great part of our knowledge of body. No man doubts of the power of aqua regia to dissolve gold, though perhaps he never himself made the experiment. In these therefore and such like cases we judge of the facts and operations of nature, upon the mere ground of testimony. However, as we can always have recourse to experience where any doubt or scruple arises, this is justly considered as the true foundation of natural philosophy; being indeed the ultimate support upon which our assent rests, and whereto we appeal when the highest degree of evidence is required.

VII. But there are many facts that will not allow of an appeal to the senses, and in this case testimony is the true and only foundation of our judgments. All human actions of whatever kind, when considered as already past, are of the nature here described; because having now no longer any existence, both the facts themselves, and the circumstances attending them, can be known only from the relations of such as had sufficient opportunities of arriving at the truth. Testimony therefore is justly accounted a third ground of human judgment; and as from the other two we have deduced scientific and natural knowledge, so we may from this derive historical; by which we mean, not merely a knowledge of the civil transactions of states and kingdoms, but of all facts whatsoever. ever, where testimony is the ultimate foundation of our belief.

CHAP. II. Of Affirmative and Negative Propositions.

I. While the comparing of our ideas is considered merely as an act of the mind, assembling them together, and joining or disjoining them according to the result of its perceptions, we call it judgment; but when our judgments are put into words, they then bear the name of proposition. A proposition therefore is a sentence expressing some judgment of the mind, whereby two or more ideas are affirmed to agree or disagree. Now as our judgments include at least two ideas, one of which is affirmed or denied of the other, so must a proposition have terms answering to these ideas. The idea of which we affirm or deny, and of course the term expressing that idea, is called the subject of the proposition. The idea affirmed or denied, as also the term answering it, is called the predicate. Thus in the proposition, God is omnipotent: God is the subject, it being of him that we affirm omnipotence; and omnipotent is the predicate, because we affirm the idea expressed by that word to belong to God.

II. But as in propositions, ideas are either joined or disjoined; it is not enough to have terms expressing those ideas, unless we have also some words to denote their agreement or disagreement. That word in a proposition, which connects two ideas together, is called the copula; and if a negative particle be annexed, we thereby understand that the ideas are disjoined. The substantive verb is commonly made use of for the copula, as in the above-mentioned proposition, God is omnipotent; where it represents the copula, and signifies the agreement of the ideas of God and omnipotence. But if we mean to separate two ideas; then, besides the substantive verb, we must also use some particle of negation, to express this repugnance. The proposition, man is not perfect, may serve as an example of this kind, where the notion of perfection being removed from the idea of man, the negative particle not is inserted after the copula, to signify the disagreement between the subject and predicate.

III. Every proposition necessarily consists of these three parts; but then it is not alike needless that they be all severally expressed in words; because the copula is often included in the term of the predicate, as when we say, he sits; which imports the same as he is sitting. In the Latin language, a single word has often the force of a whole sentence. Thus ambulat is the same as ille est ambulator; and, as ego sum amans, and so in innumerable other instances; by which it appears, that we are not so much to regard the number of words in a sentence, as the ideas they represent, and the manner in which they are put together. For wherever two ideas are joined or disjoined in an expression, though of but a single word, it is evident that we have a subject, predicate, and copula, and of consequence a complete proposition.

IV. When the mind joins two ideas, we call it an affirmative judgment; when it separates them, a negative; and as any two ideas compared together must necessarily either agree or not agree, it is evident, that all our judgments fall under these two divisions. Hence likewise the propositions expressing these judgments are all either affirmative or negative. An affirmative proposition connects the predicate with the subject, as a stone is heavy; a negative proposition separates them, as God is not the author of evil. Affirmation therefore is the same as joining two ideas together, and this is done by means of the copula. Negation on the contrary marks a repugnance between the ideas compared, in which case a negative particle must be called in, to shew that the connection included in the copula does not take place.

V. Hence we see the reason of the rule commonly laid down by logicians; that in all negative propositions the negation ought to affect the copula. For as the copula, when placed by itself, between the subject and the predicate, manifestly binds them together; it is evident, that, in order to render a proposition negative, the particles of negation must enter it in such a manner as to destroy this union. In a word, then only are two ideas disjoined in a proposition, when the negative particle may be so referred to the copula, as to break the affirmation included in it, and undo that connection it would otherwise establish. When we say, for instance, No man is perfect; take away the negation, and the copula of itself plainly unites the ideas in the proposition. But as this is the very reverse of what is intended, a negative mark is added, to shew that this union does not here take place. The negation therefore, by destroying the effect of the copula, changes the very nature of the proposition, inasmuch that, instead of binding two ideas together, it denotes their separation. On the contrary, in this sentence: The man who departs not from an upright behaviour, is beloved of God; the predicate beloved of God is evidently affirmed of the subject an upright man; so that, notwithstanding the negative particle, the proposition is still affirmative. The reason is plain; the negation here affects not the copula, but making properly a part of the subject, serves with other terms in the sentence, to form one complex idea, of which the predicate beloved of God, is directly affirmed.

CHAP. III. Of Universal and Particular Propositions.

I. The next considerable division of propositions is into universal and particular. Our ideas, according to what has been already observed in the first Part, are all singular as they enter the mind, and represent individual objects. But as by abstraction we can render them universal, so as to comprehend a whole class of things, and sometimes several classes at once; hence the terms expressing these ideas must be in like manner universal. If therefore we suppose any general term to become the subject of a proposition, it is evident, that whatever is affirmed of the abstract idea belonging to that term may be affirmed of all the individuals to which that idea extends. Thus when we say, Men are mortal; we consider mortality, not as confined to one or any number of particular men, but as what may be affirmed without restriction of the whole species. By this means the proposition becomes as general as the idea which makes the subject of it, and indeed derives its universality entirely from that idea, being more or less so according as this may be extended. extended to more or fewer individuals. But it is further to be observed of these general terms, that they sometimes enter a proposition in their full latitude, as in the example given above; and sometimes appear with a mark of limitation. In this last case we are given to understand, that the predicate agrees not to the whole universal idea, but only to a part of it; as in the proposition, *some men are wise*: For here wisdom is not affirmed of every particular man, but restrained to a few of the human species.

II. Now from this different appearance of the general idea, that constitutes the subject of any judgment, arises the division of propositions into universal and particular. An universal proposition is that wherein the subject is some general term taken in its full latitude, inasmuch that the predicate agrees to all the individuals comprehended under it, if it denotes a proper species; and to all the several species, and their individuals, if it marks an idea of a higher order. The words *all*, *every*, *no*, *none*, &c., are the proper signs of this universality; and as they seldom fail to accompany general truths, so they are the most obvious criterion whereby to distinguish them. *All animals have a power of beginning motion*. This is an universal proposition; as we know from the word *all* prefixed to the subject *animal*, which denotes that it must be taken in its full extent. Hence the power of beginning motion may be affirmed of all the several species of animals.

III. A particular proposition has in like manner some general term for its subject, but with a mark of limitation added, to denote, that the predicate agrees only to some of the individuals comprehended under a species, or to one or more of the species belonging to any genus, and not to the whole universal idea. Thus, *Some stones are heavier than iron; some men have an uncommon share of prudence*. In the last of these propositions, the subject *some men* implies only a certain number of individuals, comprehended under a single species. In the former, where the subject is a genus that extends to a great variety of distinct classes, *some stones* may not only imply any number of particular stones, but also several whole species of stones; inasmuch as there may be not a few with the property there described. Hence we see that a proposition does not cease to be particular by the predicate's agreeing to a whole species, unless that species, singly and distinctly considered, makes also the subject of which we affirm or deny.

IV. There is still one species of propositions that remains to be described, and which the more deserves our notice, as it is not yet agreed among logicians to which of the two classes mentioned above they ought to be referred; namely, singular propositions, or those where the subject is an individual. Of this nature are the following: *Sir Isaac Newton was the inventor of fluxions; this book contains many useful truths*. What occasions some difficulty as to the proper rank of these propositions is, that the subject being taken according to the whole of its extension, they sometimes have the same effect in reasoning as universals. But if it be considered that they are in truth the most limited kind of particular propositions, and that no proposition can with any propriety be called universal but where the subject is some universal idea; we shall not be long in determining to which class they ought to be referred. When we say, *Some books contain useful truths*; the proposition is particular, because the general term appears with a mark of restriction. If therefore we say, *This book contains useful truths*; it is evident that the proposition must be still more particular, as the limitation implied in the word *this* is of a more confined nature than in the former case.

V. We see therefore that all propositions are either affirmative or negative; nor is it less evident, that in both cases they may be universal or particular. Hence arises that celebrated fourfold division of them into universal affirmative and universal negative, particular affirmative and particular negative, which comprehends indeed all their varieties. The use of this method of distinguishing them will appear more fully afterwards, when we come to treat of reasoning and syllogism.

CHAP. IV. Of Absolute and Conditional Propositions.

I. The objects about which we are chiefly conversant in this world, are all of a nature liable to change. What may be affirmed of them at one time cannot often at another; and it makes no small part of our knowledge to distinguish rightly these variations, and trace the reasons upon which they depend. For it is observable, that amidst all the vicissitude of nature, some things remain constant and invariable; nor even are the changes to which we see others liable, effected, but in consequence of uniform and steady laws, which, when known, are sufficient to direct us in our judgments about them. Hence philosophers, in distinguishing the objects of our perception into various classes, have been very careful to note, that some properties belong essentially to the general idea, so as not to be separable from it but by destroying its very nature; while others are only accidental, and may be affirmed or denied of it in different circumstances. Thus solidity, a yellow colour, and great weight are considered as essential qualities of gold; but whether it shall exist as an uniform conjoined mass, is not alike necessary. We see that by a proper menstruum it may be reduced to a fine powder, and that an intense heat will bring it into a state of fusion.

II. From this diversity in the several qualities of things arises a considerable difference as to the manner of our judging about them. For all such properties as are inseparable from objects, when considered as belonging to any genus or species, are affirmed absolutely, and without reserve of that general idea. Thus we say; *Gold is very weighty, a stone is hard, animals have a power of self-motion*. But in the case of mutable or accidental qualities, as they depend upon some other consideration distinct from the general idea; that also must be taken into the account, in order to form an accurate judgment. Should we affirm, for instance, of some stones, that they are very susceptible of a rolling motion; the proposition, while it remains in this general form, cannot with any advantage be introduced into our reasonings. An attempt to receive that mode of motion flows from the figure of the stone; which, as it may vary infinitely, our judgment then only becomes applicable and determinate, nate, when the particular figure, of which volubility is a consequence, is also taken into the account. Let us then bring in this other consideration, and the proposition will run as follows: Stones of a spherical form are easily put into a rolling motion. Here we see the condition upon which the predicate is affirmed, and therefore know in what particular cases the proposition may be applied.

III. This consideration of propositions respecting the manner in which the predicate is affirmed of the subject gives rise to the division of them into absolute and conditional. Absolute propositions are those where-in we affirm some property inseparable from the idea of the subject, and which therefore belongs to it in all possible cases; as God is infinitely wise. Virtue tends to the ultimate happiness of man. But where the predicate is not necessarily connected with the idea of the subject, unless upon some consideration distinct from that idea, there the proposition is called conditional. The reason of the name is taken from the supposition annexed, which is of the nature of a condition, and may be expressed as such. Thus, If a stone is exposed to the rays of the sun, it will contract some degree of heat. If a river runs in a very declining channel, its rapidity will constantly increase.

IV. There is not any thing of greater importance in philosophy than a due attention to this division of propositions. If we are careful never to affirm things absolutely but where the ideas are inseparably joined; and if in our other judgments we distinctly mark the conditions which determine the predicate to belong to the subject; we shall be less liable to mistake in applying general truths to the particular concerns of human life. It is owing to the exact observance of this rule that mathematicians have been so happy in their discoveries, and that what they demonstrate of magnitude in general may be applied with ease in all obvious occurrences.

V. The truth of it is, particular propositions are then known to be true, when we can trace their connection with universals; and it is accordingly the great business of science to find out general truths that may be applied with safety in all obvious instances. Now the great advantage arising from determining with care the conditions upon which one idea may be affirmed or denied of another is this: that thereby particular propositions really become universal, may be introduced with certainty into our reasonings, and serve as standards to conduct and regulate our judgments. To illustrate this by a familiar instance: if we say, Some water acts very forcibly; the proposition is particular: and as the conditions on which this forcible action depends are not mentioned, it is as yet uncertain in what cases it may be applied. Let us then supply these conditions, and the proposition will run thus: Water conveyed in sufficient quantity along a steep descent acts very forcibly. Here we have an universal judgment, inasmuch as the predicate forcible action may be ascribed to all water under the circumstances mentioned. Nor is it less evident that the proposition in this new form is of easy application; and in fact we find that men do apply it in instances where the forcible action of water is required; as in corn-mills and many other works of art.

CHAP. V. Of Simple and Compound Propositions.

I. HITHERTO we have treated of propositions, Division of where only two ideas are compared together. These propositions are in the general called simple; because, having but one subject and one predicate, they are the effect of a simple judgment that admits of no subdivision. But if it so happens that several ideas offer themselves to our thoughts at once, whereby we are led to affirm the same thing of different objects, or different things of the same object; the propositions expressing these judgments are called compound: because they may be resolved into as many others as there are subjects or predicates in the whole complex determination of the mind. Thus, God is infinitely wise and infinitely powerful. Here there are two predicates, infinite wisdom and infinite power, both affirmed of the same subject; and accordingly the proposition may be resolved into two others, affirming these predicates severally. In like manner in the proposition, Neither kings nor people are exempt from death; the predicate is denied of both subjects, and may therefore be separated from them in distinct propositions. Nor is it less evident that if a complex judgment consists of several subjects and predicates, it may be resolved into as many simple propositions as are the number of different ideas compared together. Riches and honours are apt to elate the mind, and increase the number of our desires. In this judgment there are two subjects and two predicates, and it is at the same time apparent that it may be resolved into four distinct propositions. Riches are apt to elate the mind. Riches are apt to increase the number of our desires. And so of honours.

II. Logicians have divided these compound propositions into a great many different classes; but, in our opinion, not with a due regard to their proper definition. Thus conditionals, causals, relatives, &c., are mentioned as so many distinct species of this kind, though in fact they are no more than simple propositions. To give an instance of a conditional; If a stone is exposed to the rays of the sun, it will contract some degree of heat. Here we have but one subject and one predicate; for the complex expression, A stone exposed to the rays of the sun, constitutes the proper subject of this proposition, and is no more than one determinate idea. The same thing happens in causals. Rehoboam was unhappy because he followed evil counsel. There is here an appearance of two propositions arising from the complexity of the expression; but when we come to consider the matter more nearly, it is evident that we have but a single subject and predicate. The pursuit of evil counsel brought misery upon Rehoboam. It is not enough therefore to render a proposition compound, that the subject and predicate are complex notions, requiring sometimes a whole sentence to express them: for in this case the comparison is still confined to two ideas, and constitutes what we call a simple judgment. But where there are several subjects or predicates, or both, as the affirmation or negation may be alike extended to them all, the proposition expressing such a judgment is truly a collection of as many simple ones as there are different ideas compared. Confining ourselves therefore to this more strict and just notion of compound propositions, they are all reducible to two kinds, viz. copulatives and disjunctives.

III. A copulative proposition is, where the subjects and predicates are so linked together, that they may be all severally affirmed or denied one of another. Of this nature are the examples of compound propositions given above. Riches and honours are apt to elate the mind, and increase the number of our desires. Neither kings nor people are exempt from death. In the first of these the two predicates may be affirmed severally of each subject, whence we have four distinct propositions. The other furnishes an example of the negative kind, where the same predicate, being disjoined from both subjects, may be also denied of them in separate propositions.

IV. The other species of compound propositions are those called disjunctives; in which, comparing several predicates with the same subject; we affirm that one of them necessarily belongs to it, but leave the particular predicate undetermined. If any one for example says, This world either exists of itself, or is the work of some all-wise and powerful cause, it is evident that one of the two predicates must belong to the world; but as the proposition determines not which, it is therefore of the kind we call disjunctive. Such too are the following: The sun either moves round the earth, or is the centre about which the earth revolves. Friendship finds men equal, or makes them so. It is the nature of all propositions of this class, supposing them to be exact in point of form, that upon determining the particular predicate, the rest are of course to be removed; or if all the predicates but one are removed, that one necessarily takes place. Thus in the example given above; if we allow the world to be the work of some wise and powerful cause, we of course deny it to be self-existent; or if we deny it to be self-existent, we must necessarily admit that it was produced by some wise and powerful cause. Now this particular manner of linking the predicates together, so that the establishing one displaces all the rest; or the excluding all but one necessarily establishes that one; cannot otherwise be effected than by means of disjunctive particles. And hence it is that propositions of this class take their names from these particles which make no necessary a part of them, and indeed constitute their very nature considered as a distinct species.

CHAP. VI. Of the Division of Propositions into Self-evident and Demonstrable.

I. When any proposition is offered to the view of the mind, if the terms in which it is expressed and understood; upon comparing the ideas together, the agreement or disagreement asserted is either immediately perceived, or found to lie beyond the present reach of the understanding. In the first case the proposition is said to be self-evident, and admits not of any proof, because a bare attention to the ideas themselves produces full conviction and certainty; nor is it possible to call in any thing more evident by way of confirmation. But where the connection or repugnance comes not so readily under the inspection of the mind, there we must have recourse to reasoning; and if by a clear series of proofs we can make out the truth proposed, insomuch that self-evidence shall accompany every step of the procedure, we are then able to demonstrate what we assert, and the proposition itself is said to be demonstrable. When we affirm, for instance, that it is impossible for the same thing to be and not to be; whoever understands the terms made use of perceives at first glance the truth of what is asserted, nor can he by any efforts bring himself to believe the contrary. The proposition therefore is self-evident, and such that it is impossible by reasoning to make it plainer; because there is no truth more obvious or better known, from which as a consequence it may be deduced. But if we say, This world had a beginning; the assertion is indeed equally true, but shines not forth with the same degree of evidence. We find great difficulty in conceiving how the world could be made out of nothing; and are not brought to a free and full content, until by reasoning we arrive at a clear view of the absurdity involved in the contrary supposition. Hence this proposition is of the kind we call demonstrable, insomuch as its truth is not immediately perceived by the mind, but yet may be made appear by means of others more known and obvious; whence it follows as an unavoidable consequence.

II. From what has been said it appears, that reasoning is employed only about demonstrable propositions, and that our intuitive and self-evident perceptions, are the ultimate foundation on which it rests.

III. Self-evident propositions furnish the first principles of reasoning; and it is certain, that if in our researches we employ only such principles as have this character of self-evidence, and apply them according to the rules to be afterwards explained, we shall be in no danger of error in advancing from one discovery to another. For this I may appeal to the writings of the mathematicians, which, being conducted by the express model here mentioned, are an incontrovertible proof of the firmness and stability of human knowledge, when built upon so sure a foundation. For not only have the propositions of this science stood the test of ages; but are found attended with that invincible evidence, as forces the assent of all who duly consider the proofs upon which they are established.

IV. First then it is to be observed, that they have been very careful in ascertaining their ideas, and fixing the signification of their terms. For this purpose help to they begin with definitions, in which the meaning of their words is so distinctly explained, that they cannot fail to excite in the mind of an attentive reader the very same ideas as are annexed to them by the writer. And indeed the clearness and irresistible evidence of mathematical knowledge is owing to nothing so much as this care in laying the foundation. Where the relation between any two ideas is accurately and justly traced, it will not be difficult for another to comprehend that relation, if in setting himself to discover it he brings the very same ideas into comparison. But if, on the contrary, he affixes to his words ideas different from those that were in the mind of him who first advanced the demonstration; it is evident, that as the same ideas are not compared, the same relation cannot subsist, insomuch that a proposition will be rejected. rejected as false, which had the terms been rightly understood, must have appeared unexceptionably true.

A square, for instance, is a figure bounded by four equal right lines, joined together at right angles. Here the nature of the angles makes no less a part of the idea, than the equality of the sides; and many properties demonstrated of the square flow entirely from its being a rectangular figure. If therefore we suppose a man, who has formed a partial notion of a square, comprehending only the equality of its sides, without regard to the angles, reading some demonstration that implies also this latter consideration; it is plain he would reject it as not universally true, inasmuch as it could not be applied where the sides were joined together at unequal angles. For this last figure, answering still to his idea of a square, would be yet found without the property assigned to it in the proposition. But if it comes afterwards to correct his notion, and render his idea compleat, he will then readily own the truth and justness of the demonstration.

V. We see therefore, that nothing contributes so much to the improvement and certainty of human knowledge, as the having determinate ideas, and keeping them steady and invariable in all our discourses and reasonings about them. And on this account it is, that mathematicians, as was before observed, always begin by defining their terms, and distinctly unfolding the notions they are intended to express. Hence such as apply themselves to these studies have exactly the same views of things; and, bringing always the very same ideas into comparison, readily discerns the relations between them.

VI. When they have taken this first step, and made known the ideas whose relations they intend to investigate; their next care is, to lay down some self-evident truths, which may serve as a foundation for their future reasonings. And here indeed they proceed with remarkable circumspection, admitting no principles but what flow immediately from their definitions, and necessarily force themselves upon a mind in any degree attentive to its perceptions. Thus a circle is a figure formed by a right line moving round some fixed point in the same plane. The fixed point round which the line is supposed to move, and where one of its extremities terminates, is called the centre of the circle. The other extremity, which is conceived to be carried round until it returns to the point whence it first set out, describes a curve running into itself, and termed the circumference. All right lines drawn from the centre to the circumference are called radii. From these definitions compared, geometricians derive this self-evident truth: that the radii of the same circle are all equal to one another.

VII. We now observe, that in all propositions we either affirm or deny some property of the idea that constitutes the subject of our judgment, or we maintain that something may be done or effected. The first sort are called speculative propositions, as in the example mentioned above, the radii of the same circle are all equal one to another. The others are called practical, for a reason too obvious to be mentioned; thus, that a right line may be drawn from one point to another, is a practical proposition; inasmuch as it expresses that something may be done.

VIII. From this twofold consideration of propositions arises the twofold division of mathematical principles into axioms and postulates. By an axiom they understand any self-evident speculative truth; as, that the whole is greater than its parts: That things equal to one and the same thing are equal to one another. But a self-evident practical proposition is what they call a postulate. Such are those of Euclid; that a finite right line may be continued directly forwards; that a circle may be described about any centre with any distance. And here we are to observe, that as in an axiom the agreement or disagreement between the subject and predicate must come under the immediate inspection of the mind; so in a postulate, not only the possibility of the thing asserted must be evident at first view, but also the manner in which it may be effected. For where this manner is not of itself apparent, the proposition comes under the notion of the demonstrable kind, and is treated as such by geometrical writers. Thus, to draw a right-line from one point to another, is assumed by Euclid as a postulate, because the manner of doing it is so obvious, as to require no previous teaching. But then it is not equally evident, how we are to construct an equilateral triangle. For this reason he advances it as a demonstrable proposition, lays down rules for the exact performance, and at the same time proves, that if these rules are followed, the figure will be justly described.

IX. This leads us to take notice, that as self-evident and demonstrable truths are distinguished into different kinds, according as they are speculative or practical; so is it also with demonstrable propositions. A demonstrable speculative proposition is by mathematicians called a theorem. Such is the famous 47th proposition of the first book of the elements, known by the name of the Pythagorean theorem, from its supposed inventor Pythagoras, viz., "that in every right-angled triangle, the square described upon the side subtending the right-angle is equal to both the squares described upon the sides containing the right-angle." On the other hand, a demonstrable practical proposition is called a problem; as where Euclid teaches us to describe a square upon a given right-line.

X. It may not be amiss to add, that, besides the four kinds of propositions already mentioned, mathematicians have also a fifth, known by the name of corollaries. These are usually subjoined to theorems or problems, and differ from them only in this; that they flow from what is there demonstrated in so obvious a manner as to discover their dependence upon the proposition whence they are deduced, almost as soon as proposed. Thus Euclid having demonstrated, "that in every right-lined triangle all the three angles taken together are equal to two right-angles," adds by way of corollary, "that all the three angles of any one triangle taken together are equal to all the three angles of any other triangle taken together;" which is evident at first sight; because in all cases they are equal to two right ones, and things equal to one and the same thing are equal to one another.

The scholia of mathematicians are indifferently annexed to definitions, propositions, or corollaries; and answer the same purposes as annotations upon a classic or a comment. author. For in them occasion is taken to explain whatever may appear intricate and obscure in a train of reasoning; to answer objections; to reach the application and uses of propositions; to lay open the original and history of the several discoveries made in the science; and in a word, to acquaint us with all such particulars as deserve to be known, whether considered as points of curiosity or profit.

PART III. OF REASONING.

CHAP. I. Of Reasoning in general, and the parts of which it consists.

It often happens in comparing ideas together, that their agreement or disagreement cannot be discerned at first view, especially if they are of such a nature as not to admit of an exact application one to another. When, for instance, we compare two figures of a different make, in order to judge of their equality or inequality, it is plain, that by barely considering the figures themselves, we cannot arrive at an exact determination; because, by reason of their disagreeing forms, it is impossible so to put them together, as that their several parts shall mutually coincide. Here then it becomes necessary to look out for some third idea that will admit of such an application as the present case requires; wherein if we succeed, all difficulties vanish, and the relation we are in quest of may be traced with ease. Thus right-lined figures are all reduced to squares, by means of which we can measure their areas, and determine exactly their agreement or disagreement in point of magnitude.

II. But how can any third idea serve to discover a relation between two others? The answer is, by being compared severally with these others; for such a comparison enables us to see how far the ideas with which this third is compared are connected or disjoined between themselves. In the example mentioned above of two right-lined figures, if we compare each of them with some square whose area is known, and find the one exactly equal to it, and the other less by a square inch, we immediately conclude that the area of the first figure is a square inch greater than that of the second. This manner of determining the relation between any two ideas, by the intervention of some third with which they may be compared, is that which we call reasoning; and is indeed the chief instrument by which we push on our discoveries, and enlarge our knowledge. The great art lies in finding out such intermediate ideas, as, when compared with the others in the question, will furnish evident and known truths; because, as will afterwards appear, it is only by means of them that we arrive at the knowledge of what is hidden and remote.

III. Hence it appears, that every act of reasoning necessarily includes three distinct judgments; two wherein the ideas whose relation we want to discover are severally compared with the middle idea, and a third wherein they are themselves connected or disjoined, according to the result of that comparison. Now as in the second part of logic our judgments, when put into words, were called propositions, so here in the third part the expressions of our reasonings are termed syllogism. And hence it follows, that as every act of reasoning implies three several judgments, so every syllogism must include three distinct propositions. When a reasoning is thus put into words, and appears in form of a syllogism, the intermediate idea made use of, to discover the agreement or disagreement we search for, is called the middle term; and the two ideas themselves, with which this third is compared, go by the name of the extremes.

IV. But as these things are best illustrated by examples; let us, for instance, set ourselves to inquire man and see whether men are accountable for their actions. As the accountability between the ideas of man and accountability itself comes not within the immediate view of the mind, our first care must be to find out some third idea that will enable us the more easily to discover and trace it. A very small measure of reflection is sufficient to inform us, that no creature can be accountable for his actions, unless we suppose him capable of distinguishing the good from the bad; that is, unless we suppose him possessed of reason. Nor is this alone sufficient. For what would it avail him to know good from bad actions, if he had no freedom of choice, nor could avoid the one and pursue the other? hence it becomes necessary to take in both considerations in the present case. It is at the same time equally apparent, that where ever there is this ability of distinguishing good from bad actions, and of pursuing the one and avoiding the other, there also a creature is accountable. We have then got a third idea, with which accountability is inseparably connected, viz. reason and liberty; which are here to be considered as making up one complex conception. Let us now take this middle idea, and compare it with the other term in the question, viz. man, and we all know by experience that it may be affirmed of him. Having thus by means of the intermediate idea formed two several judgments, viz. that man is possessed of reason and liberty; and that reason and liberty imply accountability; a third obviously and necessarily follows, viz. that man is accountable for his actions. Here then we have a complete act of reasoning, in which, according to what has been already observed, there are three distinct judgments; two that may be filled previous, inasmuch as they lead to the other, and arise from comparing the middle idea with the two ideas in the question: the third is a consequence of these previous acts, and flows from combining the extreme ideas between themselves. If now we put this reasoning into words, it exhibits what logicians term a syllogism; and, when propounded in due form, runs thus:

"Every creature possessed of reason and liberty is accountable for his actions."

"Man is a creature possessed of reason and liberty."

"Therefore man is accountable for his actions." V. In this syllogism we may observe, that there are three several propositions expressing the three judgments implied in the act of reasoning; and so disposed, as to represent distinctly what passes within the mind in tracing the more distant relations of its ideas. The two first propositions answer the two previous judgments in reasoning, and are called the premises, because they are placed before the other. The third is termed the conclusion, as being gained in consequence of what was affected in the premises. We are also to remember, that the terms expressing the two ideas whose relations we inquire after, as here man and accountable, are in general called the extremes; and that the intermediate idea, by means of which the relation is traced, viz. a creature possessed of reason and liberty, takes the name of the middle term. Hence it follows, that by the premises of a syllogism we are always to understand the two propositions where the middle term is severally compared with the extremes; for these constitute the previous judgments, whence the truth we are in quest of is by reasoning deduced. The conclusion is that other proposition, in which the extremes themselves are joined or separated agreeably to what appears upon the above comparison.

VI. The conclusion is made up of the extreme terms of the syllogism; and the extreme, which serves as the predicate of the conclusion, goes by the name of the major term; the other extreme, which makes the subject of the same proposition, is called the minor term. From this distinction of the extremes arises also a distinction between the premises, where these extremes are severally compared with the middle term. That proposition which compares the greater extreme, or the predicate of the conclusion with the middle term, is called the major proposition; the other, wherein the same middle term is compared with the subject of the conclusion or lesser extreme, is called the minor proposition. All this is obvious from the syllogism already given, where the conclusion is, man is accountable for his actions. For here the predicate accountable for his actions, being connected with the middle term in the first of the two premises; every creature possessed of reason and liberty is accountable for his actions, gives what we call the major proposition. In the second of the premises; man is a creature possessed of reason and liberty, we find the lesser extreme, or subject of the conclusion, viz. man, connected with the same middle term, whence it is known to be the minor proposition. When a syllogism is proposed in due form, the major proposition is always placed first, the minor next, and the conclusion last.

VII. These things premised, we may in the general define reasoning to be an act or operation of the mind, deducing some unknown proposition from other previous ones that are evident and known. These previous propositions, in a simple act of reasoning, are only two in number; and it is always required that they be of themselves apparent to the understanding, inasmuch that we assent to and perceive the truth of them as soon as proposed. In the syllogism given above, the premises are supposed to be self-evident truths; otherwise the conclusion could not be inferred by a single act of reasoning. If, for instance, in the major, every creature possessed of reason and liberty is accountable for his actions, the connection between the subject and predicate could not be perceived by a bare attention to the ideas themselves; it is evident that this proposition would no less require a proof than the conclusion deduced from it. In this case a new middle term must be sought for, to trace the connection here supposed; and this of course furnishes another syllogism, by which having established the proposition in question, we are then, and not before, at liberty to use it in any succeeding train of reasoning. And should it so happen, that in this second essay there was still some previous proposition whose truth did not appear at first sight, we must then have recourse to a third syllogism, in order to lay open that truth to the mind; because so long as the premises remain uncertain, the conclusion built upon them must be so too. When, by conducting our thoughts in this manner, we at last arrive at some syllogism where the previous propositions are intuitive truths; the mind then rests in full security, as perceiving that the several conclusions it has passed through stand upon the immoveable foundation of self-evidence, and when traced to their source terminate in it.

VIII. We see therefore, that in order to infer a Reasoning, conclusion by a single act of reasoning, the premises must be intuitive propositions. Where they are not, previous syllogisms are required; in which case reasoning becomes a complicated act, taking in a variety of successive steps. This frequently happens in tracing the more remote relation of our ideas; where, many middle terms being called in, the conclusion cannot be made out but in consequence of a series of syllogisms following one another in train. But although in this concatenation of propositions, those that form the premises of the last syllogism are often considerably removed from self-evidence; yet if we trace the reasoning backwards, we shall find them the conclusions of previous syllogisms, whose premises approach nearer and nearer to intuition in proportion as we advance, and are found at last to terminate in it. And if, after having thus unravelled a demonstration, we take it the contrary way; and observe how the mind, setting out with intuitive perceptions, couples them together to form a conclusion; how, by introducing this conclusion into another syllogism, it still advances one step farther; and so proceeds, making every new discovery subservient to its future progress; we shall then perceive clearly, that reasoning, in the highest exercise of that faculty, is no more than an orderly combination of those simple acts which we have already so fully explained.

IX. Thus we see, that reasoning beginning with first principles, rises gradually from one judgment to another, and connects them in such manner, that every stage of the progression brings intuitive certainty along with it. And now at length we may clearly understand the definition given above of this distinguishing faculty of the human mind. Reason, we have said, is the ability of deducing unknown truths from principles or propositions that are already known. This evidently appears by the foregoing account, where we see that no proposition is admitted into a syllogism, to serve as one of the previous judgments on which the conclusion rests, unless it is itself a known and established. blished truth, whose connection with self-evident principles has been already traced.

**CHAP. II. Of the several kinds of Reasoning; and first, of that by which we determine the Genera and Species of Things.**

I. All the aims of human reason may in the general be reduced to these two: 1. To rank things under those universal ideas to which they truly belong; and, 2. To ascribe to them their several attributes and properties in consequence of that distribution.

II. One great aim of human reason is to determine the genera and species of things. We have seen in the first Part of this treatise, how the mind proceeds in framing general ideas. We have also seen in the second Part, how by means of these general ideas we come by universal propositions. Now as in these universal propositions we affirm some property of a genus or species, it is plain that we cannot apply this property to particular objects till we have first determined whether they are comprehended under that general idea of which the property is affirmed. Thus there are certain properties belonging to all even numbers, which nevertheless cannot be applied to any particular number, until we have first discovered it to be of the species expressed by that natural name. Hence reasoning begins with referring things to their several divisions and classes in the scale of our ideas; and as these divisions are all distinguished by particular names, we hereby learn to apply the terms expressing general conceptions to such particular objects as come under our immediate observation.

III. Now in order to arrive at these conclusions, by which the several objects of perception are brought under general names, two things are manifestly necessary. First, that we take a view of the idea itself denoted by that general name, and carefully attend to the distinguishing marks which serve to characterize it. Secondly, that we compare this idea with the object under consideration, observing diligently wherein they agree or differ. If the idea is found to correspond with the particular object, we then without hesitation apply the general name; but if no such correspondence intervenes, the conclusion must necessarily take a contrary turn. Let us, for instance, take the number eight, and consider by what steps we are led to pronounce it an even number. First then, we call to mind the idea signified by the expression an even number, viz. that it is a number divisible into two equal parts. We then compare this idea with the number eight, and, finding them manifestly to agree, see at once the necessity of admitting the conclusion. These several judgments therefore transferred into language, and reduced to the form of a syllogism, appear thus:

"Every number that may be divided into two equal parts is an even number."

"The number eight may be divided into two equal parts."

"Therefore the number eight is an even number."

IV. Here it may be observed, that where the general idea, to which particular objects are referred, is very familiar to the mind, and frequently in view; this reference, and the application of the general name, seem to be made without any apparatus of reasoning. When we see a horse in the fields, or a dog in the street, we readily apply the name of the species; habit, and a familiar acquaintance with the general idea, suggesting it instantaneously to the mind. We are not however to imagine on this account that the understanding departs from the usual rules of just thinking. A frequent repetition of acts begets a habit; and habits are attended with a certain promptness of execution, that prevents our observing the several steps and gradations by which any course of action is accomplished. But in other instances, where we judge not by precontracted habits, as when the general idea is very complex, or less familiar to the mind, we always proceed according to the form of reasoning established above. A goldsmith, for instance, who is in doubt as to any piece of metal, whether it be of the species called gold, first examines its properties, and then comparing them with the general idea signified by that name, if he finds a perfect correspondence, no longer hesitates under what class of metals to rank it.

V. Nor let it be imagined that our researches here, The great because in appearance bounded to the imposing of general names upon particular objects, are therefore trivial and of little consequence. Some of the most considerable debates among mankind, and such too as nearly regard their lives, interest, and happiness, turn wholly upon this article. Is it not the chief employment of our several courts of judicature to determine in particular instances, what is law, justice, and equity? Of what importance is it in many cases to decide a right whether an action shall be termed murder or manslaughter? We see then that no less than the lives and fortunes of men depend often upon these decisions. The reason is plain. Actions, when once referred to a general idea, draw after them all that may be affirmed of that idea; insomuch that the determining the species of actions is all one with determining what proportion of praise or dispraise, commendation or blame, &c. ought to follow them. For as it is allowed that murder deserves death; by bringing any particular action under the head of murder, we of course decide the punishment due to it.

VI. But the great importance of this branch of reasoning, and the necessity of care and circumspection in referring particular objects to general ideas, is still farther evident from the practice of the mathematicians. Every one who has read Euclid knows, that he frequently requires us to draw lines through certain points, and according to such and such directions. The figures thence resulting are often squares, parallelograms, or rectangles. Yet Euclid never supposes this from their bare appearance, but always demonstrates it upon the strictest principles of geometry. Nor is the method he takes in any thing different from that described above. Thus, for instance, having defined a square to be a figure bounded by four equal sides joined together at right angles; when such a figure arises in any construction previous to the demonstration of a proposition, yet he never calls it by that name until he has shewn that its sides are equal, and all its angles right ones. Now this is apparently the same form of reasoning we have before exhibited in proving eight to be an even number.

VII. Having VII. Having thus explained the rules by which we are to conduct ourselves in ranking particular objects under general ideas, and shewn their conformity to the practice and manner of the mathematicians; it remains only to observe, that the true way of rendering this part of knowledge both easy and certain, is by habituating ourselves to clear and determinate ideas, and keeping them steadily annexed to their respective names. For as all our aim is to apply general words aright, if these words stand for invariable ideas that are perfectly known to the mind, and can be readily distinguished upon occasion, there will be little danger of mistake or error in our reasonings. Let us suppose that, by examining any object, and carrying our attention successively from one part to another, we have acquainted ourselves with the several particulars observable in it. If among these we find such as constitute some general idea, framed and settled beforehand by the understanding, and distinguished by a particular name, the resemblance thus known and perceived necessarily determines the species of the object, and thereby gives it a right to the name by which that species is called. Thus four equal sides, joined together at right angles, make up the notion of a square. As this is a fixed and invariable idea, without which the general name cannot be applied, we never call any particular figure a square until it appears to have these several conditions; and contrariwise, wherever a figure is found with these conditions, it necessarily takes the name of a square. The same will be found to hold in all our other reasonings of this kind, where nothing can create any difficulty but the want of settled ideas. If, for instance, we have not determined within ourselves the precise notion denoted by the word manslaughter, it will be impossible for us to decide whether any particular action ought to bear that name: because, however nicely we examine the action itself, yet, being strangers to the general idea with which it is to be compared, we are utterly unable to judge of their agreement or disagreement. But if we take care to remove this obstacle, and distinctly trace the two ideas under consideration, all difficulties vanish, and the resolution becomes both easy and certain.

VIII. Thus we see of what importance it is towards the improvement and certainty of human knowledge, that we accustom ourselves to clear and determinate ideas, and a steady application of words.

CHAP. III. Of Reasoning, as it regards the Powers and Properties of Things, and the Relations of our general Ideas.

I. We come now to the second great end which men have in view in their reasonings; namely, the discovering and ascribing to things their several attributes and properties. And here it will be necessary to distinguish between reasoning, as it regards the sciences, and as it concerns common life. In the sciences, our reason is employed chiefly about universal truths, it being by them alone that the bounds of human knowledge are enlarged. Hence the division of things into various classes, called otherwise genera and species. For these universal ideas being set up as the representatives of many particular things, whatever is affirmed of them may be also affirmed of all the individuals to which they belong. Murder, for instance, is a general idea, representing a certain species of human actions. Reason tells us that the punishment due to it is death. Hence every particular action, coming under the notion of murder, has the punishment of death allotted to it. Here then we apply the general truth to some obvious instance; and this is what properly constitutes the reasoning of common life. For men, in their ordinary transactions and intercourse one with another, have, for the most part, to do only with particular objects. Our friends and relations, their characters and behaviour, the constitution of the several bodies that surround us, and the uses to which they may be applied, are what chiefly engage our attention. In all these, we reason about particular things; and the whole result of our reasoning is, the applying the general truths of the sciences in the ordinary transactions of human life. When we see a viper, we avoid it. Wherever we have occasion for the forcible action of water to move a body that makes considerable resistance, we take care to convey it in such a manner that it shall fall upon the object with impetuosity. Now all this happens in consequence of our familiar and ready application of these two general truths. The bite of a viper is mortal. Water, falling upon a body with impetuosity, acts very forcibly towards setting it in motion. In like manner, if we let ourselves to consider any particular character, in order to determine the share of praise or dispraise that belongs to it, our great concern is to ascertain exactly the proportion of virtue and vice. The reason is obvious. A just determination, in all cases of this kind, depends entirely upon an application of these general maxims of morality. Virtuous actions deserve praise. Vicious actions deserve blame.

II. Hence it appears that reasoning, as it regards common life, is no more than the ascribing the general properties of things to those several objects with which we are more immediately concerned, according as they are found to be of that particular division or class to which the properties belong. The steps then by which we proceed are manifestly these. First, we refer the object under consideration to some general idea or class of things. We then recollect the several attributes of that general idea. And, lastly, ascribe all those attributes to the present object. Thus, in considering the character of Sempronius, if we find it to be of the kind called virtuous, when we at the same time reflect that a virtuous character is deserving of esteem, it naturally and obviously follows that Sempronius is so too. These thoughts put into a syllogism, in order to exhibit the form of reasoning here required, run thus:

"Every virtuous man is worthy of esteem. "Sempronius is a virtuous man: "Therefore Sempronius is worthy of esteem."

III. By this syllogism it appears, that before we affirm any thing of a particular object, that object must be referred to some general idea. Sempronius is pronounced worthy of esteem only in consequence of his being a virtuous man, or coming under that general notion. Hence we see the necessary connection of the various parts of reasoning, and the dependence they they have one upon another. The determining the genera and species of things is, as we have said, one exercise of human reason; and here we find that this exercise is the first in order, and previous to the other, which consists in ascribing to them their powers, properties, and relations. But when we have taken this previous step, and brought particular objects under general names; as the properties we ascribe to them are no other than those of the general idea, it is plain that, in order to a successful progress in this part of knowledge, we must thoroughly acquaint ourselves with the several relations and attributes of these our general ideas. When this is done, the other part will be easy, and requires scarce any labour or thought, as being no more than an application of the general form of reasoning represented in the foregoing syllogism. Now as we have already sufficiently shown how we are to proceed in determining the genera and species of things, which, as we have said, is the previous step to this second branch of human knowledge; all that is farther wanting towards a due explanation of it is, to offer some considerations as to the manner of investigating the general relations of our ideas. This is the highest exercise of the powers of the understanding, and that by means whereof we arrive at the discovery of universal truths; inasmuch that our deductions in this way constitute that particular species of reasoning which we have before said regards principally the sciences.

IV. But that we may conduct our thoughts with some order and method, we shall begin with observations, that the relations of our general ideas are of two kinds: either such as immediately discover themselves, upon comparing the ideas one with another; or such as, being more remote and distant, require art and contrivance to bring them into view. The relations of the first kind furnish us with intuitive and self-evident truths; those of the second are traced by reasoning, and a due application of intermediate ideas. It is of this last kind that we are to speak here, having dispatched what was necessary with regard to the other in the second Part. As, therefore, in tracing the more distant relations of things, we must always have recourse to intervening ideas, and are more or less successful in our researches according to our acquaintance with these ideas, and ability of applying them; it is evident that, to make a good reasoner, two things are principally required. First, An extensive knowledge of those intermediate ideas, by means of which things may be compared one with another. Secondly, The skill and talent of applying them happily in all particular instances that come under consideration.

V. In order to our successful progress in reasoning, we must have an extensive knowledge of those intermediate ideas by means of which things may be compared one with another. For as it is not every idea that will answer the purpose of our inquiries, but such only as are peculiarly related to the objects about which we reason, so as, by a comparison with them, to furnish evident and known truths; nothing is more apparent than that the greater variety of conceptions we can call into view, the more likely we are to find some among them that will help us to the truths here required. And, indeed, it is found to hold in experience, that in proportion as we enlarge our views of things, and grow acquainted with a multitude of different objects, the reasoning faculty gathers strength; for, by extending our sphere of knowledge, the mind acquires a certain force and penetration, as being accustomed to examine the several appearances of its ideas, and observe what light they cast one upon another.

VI. This is the reason why, in order to excel remarkably in any one branch of learning, it is necessary to have at least a general acquaintance with the whole circle of arts and sciences. The truth of it is, all the various divisions of human knowledge are very nearly related among themselves, and, in innumerable instances, serve to illustrate and set off each other. And although it is not to be denied that, by an obstinate application to one branch of study, a man may make considerable progress, and acquire some degree of eminence in it; yet his views will be always narrow and contracted, and he will want that masterly discernment which not only enables us to pursue our discoveries with ease, but also, in laying them open to others, to spread a certain brightness around them. But when our reasoning regards a particular science, it is farther necessary that we more nearly acquaint ourselves with whatever relates to that science. A general knowledge is a good preparation, and enables us to proceed with ease and expedition in whatever branch of learning we apply to. But then, in the minute and intricate questions of any science, we are by no means qualified to reason with advantage until we have perfectly mastered the science to which they belong.

VII. We come now to the second thing required, in order to a successful progress in reasoning; namely, the skill and talent of applying intermediate ideas happily in all particular instances that come under consideration. And here, rules and precepts are of little service. Use and experience are the best instructors. For, whatever logicians may boast of being able to form perfect reasoners by book and rule, we find by experience, that the study of their precepts does not always add any great degree of strength to the understanding. In short, it is the habit alone of reasoning that makes a reasoner. And therefore the true way to acquire this talent is, by being much conversant in those sciences where the art of reasoning is allowed to reign in the greatest perfection. Hence it was that the ancients, who so well understood the manner of forming the mind, always began with mathematics, as the foundation of their philosophical studies. Here the understanding is by degrees habituated to truth, contracts intensively a certain fondness for it, and learns never to yield its assent to any proposition, but where the evidence is sufficient to produce full conviction. For this reason Plato has called mathematical demonstrations the cathartics or purgatives of the soul, as being the proper means to cleanse it from error, and restore that natural exercise of its faculties in which just thinking consists.

VIII. If therefore we would form our minds to a habit of reasoning closely and in train, we cannot take take any more certain method, than the exercising ourselves in mathematical demonstrations, so as to contract a kind of familiarity with them. Not that we look upon it as necessary that all men should be deep mathematicians; but that, having got the way of reasoning which that study necessarily brings the mind to, they may be able to transfer it to other parts of knowledge, as they shall have occasion.

IX. But although the study of mathematics be of all others the most useful, to form the mind, and give it an early relish of truth, yet ought not other parts of philosophy to be neglected. For there also we meet with many opportunities of exercising the powers of the understanding; and the variety of subjects naturally leads us to observe all those different turns of thinking that are peculiarly adapted to the several ideas we examine, and the truth we search after. A mind thus trained acquires a certain mastery over its own thoughts, inasmuch that it can range and model them at pleasure, and call such into view as best suit its present designs. Now in this the whole art of reasoning consists; from among a great variety of different ideas to single out those that are most proper for the business in hand, and to lay them together in such order, that from plain and easy beginnings, by gentle degrees, and a continued train of evident truths, we may be insensibly led on to such discoveries, as at our first setting out appeared beyond the reach of human understanding. For this purpose, besides the study of mathematics before recommended, we ought to apply ourselves diligently to the reading of such authors as have distinguished themselves for strength of reasoning, and a just and accurate manner of thinking. For it is observable, that a mind exercised and seasoned to truth, seldom rests satisfied in a bare contemplation of the arguments offered by others; but will be frequently assaying its own strength, and pursuing its discoveries upon the plan it is most accustomed to. Thus we insensibly contract a habit of tracing truth from one stage to another, and of investigating those general relations and properties which we afterwards ascribe to particular things, according as we find them comprehended under the abstract ideas to which the properties belong.

CHAP. IV. Of the Forms of Syllogisms.

I. HITHERTO we have contented ourselves with a general notion of syllogisms, and of the parts of which they consist. It is now time to enter a little more particularly into the subject, to examine their various forms, and lay open the rules of argumentation proper to each. In the syllogisms mentioned in the foregoing chapters, we may observe, that the middle term is the subject of the major proposition, and the predicate of the minor. This disposition, though the most natural and obvious, is not however necessary; it frequently happening, that the middle term is the subject in both the premises, or the predicate in both; and sometimes, directly contrary to its disposition in the foregoing chapters, the predicate in the major, and the subject in the minor. Hence the distinction of syllogisms into various kinds, called figures by logicians. For figure, according to their use of the word, is nothing else but the order and disposition of the middle term in any syllogism. And as this disposition is, we see, fourfold, so the figures of syllogisms thence arising are four in number. When the middle term is the subject of the major proposition, and the predicate of the minor, we have what is called the first figure. If, on the other hand, it is the predicate of both the premises, the syllogism is said to be the second figure. Again, in the third figure, the middle term is the subject of the two premises. And lastly, by making it the predicate of the major, and subject of the minor, we obtain syllogisms in the fourth figure.

II. But, besides this fourfold division of syllogisms, there is also a farther subdivision of them in every figure, arising from the quantity and quality, as they are called, of the propositions. By quantity we mean the consideration of propositions, as universal or particular; by quality, as affirmative or negative.

Now as, in all the several dispositions of the middle term, the propositions of which a syllogism consists may be either universal or particular, affirmative or negative; the due determination of these, and so putting them together as the laws of argumentation require, constitute what logicians call the moods of syllogisms. Of these moods there is a determinate number to every figure, including all the possible ways in which propositions differing in quantity or quality can be combined, according to any disposition of the middle term, in order to arrive at a just conclusion.

III. The division of syllogisms according to mood and figure respects those especially which are known by the name of plain simple syllogisms; that is, which are bounded to three propositions, all simple, and where the extremes and middle term are connected, according to the rules laid down above. But as the mind is not tied down to any one precise form of reasoning, but sometimes makes use of more, sometimes of fewer premises, and often takes in compound and conditional propositions, it may not be amiss to take notice of the different forms derived from this source, and explain the rules by which the mind conducts itself in the use of them.

IV. When in any syllogism the major is a conditional proposition, the syllogism itself is termed conditional. Thus:

"If there is a God, he ought to be worshipped." "But there is a God:" "Therefore he ought to be worshipped."

In this example, the major, or first proposition, is, we see, conditional, and therefore the syllogism itself is also of the kind called by that name. And here we are to observe, that all conditional propositions are made of two distinct parts: one expressing the condition upon which the predicate agrees or disagrees with the subject, as in this now before us, if there is a God; the other joining or disjoining the said predicate and subject, as here, he ought to be worshipped. The first of these parts, or that which implies the condition, is called the antecedent; the second, where we join or disjoin the predicate and subject, has the name of the consequent.

V. In all propositions of this kind, supposing them to be exact in point of form, the relation between the antecedent and consequent must ever be true and real; that is, the antecedent must always contain some certain tain and genuine condition, which necessarily implies the consequent; for otherwise the proposition itself will be false, and therefore ought not to be admitted into our reasonings. Hence it follows, that when any conditional proposition is assumed, if we admit the antecedent of that proposition, we must at the same time necessarily admit the consequent, but if we reject the consequent, we are in like manner bound to reject the antecedent. For as the antecedent always expresses some condition which necessarily implies the truth of the consequent; by admitting the antecedent, we allow of that condition, and therefore ought also to admit the consequent. In like manner, if it appears that the consequent ought to be rejected, the antecedent evidently must be too; because, as was just now demonstrated, the admitting of the antecedent would necessarily imply the admission also of the consequent.

VI. There are two ways of arguing in hypothetical syllogisms, which lead to a certain and unavoidable conclusion. For as the major is always a conditional proposition, consisting of an antecedent and a consequent; if the minor admits the antecedent, it is plain that the conclusion must admit the consequent. This is called arguing from the admission of the antecedent to the admission of the consequent, and constitutes that mood or species of hypothetical syllogisms which is distinguished in the schools by the name of the modus ponens, inasmuch as by it the whole conditional proposition, both antecedent and consequent, is established. Thus:

"If God is infinitely wise, and acts with perfect freedom, he does nothing but what is best."

"But God is infinitely wise, and acts with perfect freedom."

"Therefore he does nothing but what is best."

Here we see the antecedent or first part of the conditional proposition is established in the minor; and the consequent or second part in the conclusion; whence the syllogism itself is an example of the modus ponens. But if now we on the contrary suppose that the minor rejects the consequent, then it is apparent that the conclusion must also reject the antecedent. In this case we are said to argue from the removal of the consequent to the removal of the antecedent, and the particular mood or species of syllogisms thence arising is called by logicians the modus tollens; because in it both antecedent and consequent are rejected or taken away, as appears by the following example:

"If God were not a Being of infinite goodness, neither would he consult the happiness of his creatures."

"But God does consult the happiness of his creatures."

"Therefore he is a Being of infinite goodness."

VII. These two species take in the whole class of conditional syllogisms, and include all the possible ways of arguing that lead to a legitimate conclusion; because we cannot here proceed by a contrary process of reasoning, that is, from the removal of the antecedent to the removal of the consequent, or from the establishing of the consequent to the establishing of the antecedent. For although the antecedent always expresses some real condition, which, once admitted, necessarily implies the consequent, yet it does not follow that there is therefore no other condition; and if so, then, after removing the antecedent, the consequent may still hold, because of some other determination that enters it. When we say, "If a stone is exposed some time to the rays of the Sun, it will contract a certain degree of heat;" the proposition is certainly true; and, admitting the antecedent, we must also admit the consequent. But as there are other ways by which a stone may gather heat, it will not follow, from the ceasing of the before-mentioned condition, that therefore the consequent cannot take place. In other words, we cannot argue: "But the stone has not been exposed to the rays of the sun; therefore neither has it any degree of heat." Inasmuch as there are a great many other ways by which heat might have been communicated to it. And if we cannot argue from the removal of the antecedent to the removal of the consequent, no more can we from the admission of the consequent to the admission of the antecedent; because, as the consequent may flow from a great variety of different suppositions, the allowing of it does not determine the precise supposition, but only that some one of them must take place. Thus in the foregoing proposition, "If a stone is exposed sometime to the rays of the sun, it will contract a certain degree of heat;" admitting the consequent, viz. that it has contracted a certain degree of heat, we are not therefore bound to admit the antecedent, that it has been sometime exposed to the rays of the sun; because there are many other causes whence that heat may have proceeded. These two ways of arguing, therefore, hold not in conditional syllogisms.

VIII. As from the major's being a conditional proposition, we obtain the species of conditional syllogisms; so where it is a disjunctive proposition, the syllogism to which it belongs, is also called disjunctive syllogisms, as in the following example:

"The world is either self-existent, or the work of some finite, or of some infinite Being."

"But it is not self-existent, nor the work of a finite being."

"Therefore it is the work of an infinite Being."

Now a disjunctive proposition is that, where of several predicates, we affirm one necessarily to belong to the subject, to the exclusion of all the rest, but leave that particular one undetermined. Hence it follows, that as soon as we determine the particular predicate, all the rest are of course to be rejected; or if we reject all the predicates but one, that one necessarily takes place. When, therefore, in a disjunctive syllogism, the several predicates are enumerated in the major; if the minor establishes any one of these predicates, the conclusion ought to remove all the rest; or if, in the minor, all the predicates but one are removed, the conclusion must necessarily establish that one. Thus, in the disjunctive syllogism given above, the major affirms one of the three predicates to belong to the earth, viz. self-existence, or that it is the work of a finite, or that it is the work of an infinite Being. Two of these predicates are removed in the minor, viz. self-existence, and the work of a finite being. Hence the conclusion necessarily attributes to it the third predicate, and affirms that it is the work of an infinite Being. If now we give the syllogism another turn, inasmuch that the minor may establish one of the predicates, predicates, by affirming the earth to be the production of an infinite Being; then the conclusion must remove the other two, afflicting it to be neither self-existent, nor the work of a finite being. These are the forms of reasoning in these species of syllogisms, the justness of which appears at first sight; and that there can be no other, is evident from the very nature of a disjunctive proposition.

IX. In the several kinds of syllogisms hitherto mentioned, we may observe, that the parts are complete; that is, the three propositions of which they consist are represented in form. But it often happens, that some one of the premises is not only an evident truth, but also familiar and in the minds of all men; in which case it is usually omitted, whereby we have an imperfect syllogism, that seems to be made up of only two propositions. Should we, for instance, argue in this manner:

"Every man is mortal: "Therefore every king is mortal."

The syllogism appears to be imperfect, as consisting but of two propositions. Yet it is really complete; only the minor [every king is a man] is omitted, and left to the reader to supply, as being a proposition so familiar and evident that it cannot escape him.

X. These seemingly imperfect syllogisms are called enthymemes, and occur very frequently in reasoning, especially where it makes a part of common conversation. Nay, there is a particular elegance in them, because, not displaying the argument in all its parts, they leave somewhat to the exercise and invention of the mind. By this means we are put upon exerting ourselves, and seem to share in the discovery of what is proposed to us. Now this is the great secret of fine writing, to frame and put together our thoughts, as to give full play to the reader's imagination, and draw him infallibly into our very views and course of reasoning. This gives a pleasure not unlike to that which the author himself feels in composing. It besides shortens discourse, and adds a certain force and liveliness to our arguments, when the words in which they are conveyed favour the natural quickness of the mind in its operations, and a single expression is left to exhibit a whole train of thoughts.

XI. But there is another species of reasoning with two propositions, which seems to be complete in itself, and where we admit the conclusion without supposing any tacit or suppressed judgment in the mind, from which it follows syllogistically. This happens between propositions, where the connection is such, that the admission of the one necessarily and at the first sight implies the admission also of the other. For if it so falls out, that the proposition on which the other depends is self-evident, we content ourselves with barely affirming it, and infer that other by a direct conclusion. Thus, by admitting an universal proposition, we are forced also to admit of all the particular propositions comprehended under it; this being the very condition that constitutes a proposition universal. If then that universal proposition chances to be self-evident, the particular ones follow of course, without any farther train of reasoning. Whoever allows, for instance, that things equal to one and the same thing are equal to one another, must at the same time allow, that two triangles, each equal to a square whose side is three inches, are also equal between themselves. This argument therefore,

"Things equal to one and the same thing, are equal to one another: "Therefore these two triangles, each equal to the square of a line of three inches, are equal between themselves."

is complete in its kind, and contains all that is necessary towards a just and legitimate conclusion. For the first or universal proposition is self-evident, and therefore requires no farther proof. And as the truth of the particular is inseparably connected with that of the universal, it follows from it by an obvious and unavoidable consequence.

XII. Now in all cases of this kind, where propositions are deduced one from another, on account of a known and evident connection, we are said to reason by immediate consequence. Such a coherence of propositions manifest at first sight, and forcing itself upon the mind, frequently occurs in reasoning. Logicians have explained at some length the several suppositions upon which it takes place, and allow of all immediate consequences that follow in conformity to them. It is however observable, that these arguments, though seemingly complete, because the conclusion follows necessarily from the single proposition that goes before, may yet be considered as real enthymemes, whose major, which is a conditional proposition, is wanting. The syllogism just mentioned, when represented according to this view, will run as follows:

"If things equal to one and the same thing, are equal to one another; these two triangles, each equal to a square whose side is three inches, are also equal between themselves.

"But things equal to one and the same thing, are equal to one another: "Therefore also these triangles, &c., are equal between themselves."

This observation will be found to hold in all immediate consequences whatsoever, inasmuch that they are in fact no more than enthymemes of hypothetical syllogisms. But then it is particular to them, that the ground on which the conclusion rests, namely its coherence with the minor, is of itself apparent, and seen immediately to flow from the rules and reasons of logic.

XIII. The next species of reasoning we shall take notice of here is what is commonly known by the name plain simple of a forites. This is a way of arguing, in which a great number of propositions are so linked together, that the predicate of one becomes continually the subject of the next following, until at last a conclusion is formed, by bringing together the subject of the first proposition, and the predicate of the last. Of this kind is the following argument:

"God is omnipotent. "An omnipotent being can do everything possible. "He that can do everything possible, can do whatever involves not a contradiction. "Therefore God can do whatever involves not a contradiction."

This particular combination of propositions may be continued to any length we please, without in the least weakening.