Suits. However, in compliance with the wishes of his friends, he applied to theological studies. When under examination for the pulpit he was required to expound one of the Psalms. The stiff and rigid Scotch theologian pronounced the version too poetical. The condemnation engendered in Thomson so great an aversion to theology, that from that moment he resolved to devote himself to the art for which he felt nature had created him. At the time there existed in the University a Belles Lettres Club, which called itself "The Athenian Society." This society publis"hed a collection of poetry under the title of "The Edinburgh Miscellany." To this work Thomson appears to have contributed, though his pieces have not been identified. Convinced that his talents could find due recognition only in London, and in the hope of obtaining some assistance from a family known to him that lived there, he betook himself to the metropolis in the autumn of 1725. In London, among other of his college mates, he iound David Mallet, who filled the post of tutor to a son of the Duke of Montrose. To him he showed his "'Winter," which then consisted simply at fragments. At Mallet's suggestion he finished and published the poem. It appeared in 1726, and had a favourable reception. It is still accounted Thomson's master-piece. The dedication of the poem to Sir Spencer Compton procured the author only twenty guineas. A larger reward, however, he found in the reputation it brought him. Among other advantages it gained him an acquaintance with Pope. In 1727 he published his "Summer,' dedicated to Lord Melcombe. In the same year he put forth a " Poem Sacred to the Memory of Sir Isaac Newton," and another under the title of "Britannia," being an invective against the government. Next appeared (1728) "Spring," with a dedication to the Countess of Hertford, afterwards Duchess ot Somerset. This compliment brought him an invitation to Lord Hertford's mansion, where he remained some months. The "Autumn" did not appear until our poet published an edition of his works in 1730. The same year Thomson placed on the stage his tragedy of " Sophonisba." Having been read in private circles, the play had excited great expectations. Disappointment ensued. The line "Oh, Sophonisba! Sophonisba, oh!" was parodied thus,! "Oh, Jemmy Thomson! Jemmy Thomson, oh!" Not long after, the poet accompanied Charles Talbot, eldest son of the Lord Chancellor, in a tour on the continent, whence he returned home enriched with varied knowledge. He was rewarded for his care of the young man with an office, which secured him an income without requiring much of his time. While on his travels he formed the first idea of his poem on "Liberty," to the composition of which he devoted two years. During the time his pupil died. The loss occasioned the dedication to the young man's memory of a few lines at the commencement. Thomson set a great value on this production, and expected it would secure unusual acceptance. The hope was but partially fulfilled. It consists of five books or cantos, having these titles: 1, Ancient and Modern Italy compared, 2. Greece, 3. Rome, 4. Britain, 5. The Prospect. A short time after its publication his patron died, and Thomson lost his place. In consequence he was thrown on literature as a source of support. In 1738 his tragedy of" Agamemnon" was submitted to the public in Drury-lane Theatre. It obtained no very marked success. About this time Thomson became acquainted with the Prince of Wales, who, affecting popularity, had become a patron of poets and scholars. He gave our poet a pension of £100. In 1731 the prince desired that Thomson's tragedy, "Edward and Elenora," should be played, but the Lord Chamberlain forbad the representation, because the author was of the party in opposition to the Court. In 1710 Thomson, in union with Mallet, produced "The Masque of Alfred," a piece which was represented in honour of the Princess Augusta on her birthday. In 1745 his tragedy of "Tancred and Sigismunda" was exhibited, and won greater popularity than any other of his plays. The last of his publications, and perhaps the loveliest, his " Castle of Indolence," appeared in 1716. About this time Thomson, through the influence of his friend Mr. Lyttleton, obtained the post of "Surveyor of the Leeward Islands," a Btrange office for a poet; but the unsuitableness was compensated for by an mcome of £300 a-year. He might now have lived at his ease, but a cold which he took soon ended his days (Aug. 27th, 1748). A tragedy, named " Coriolanus," was published after his decease. Among Thomson's works, "The Seasons" may be pronounced the best. Thomson drew his pictures immediately from nature. Hence their truth, their lively colouring and their varied beauty. He says of himself in his "Autumn," "I solitary court Th' inspiring breeze, and meditate the book 'Of Nature, ever open; aiming thence Warm from the heart to pour the moral song." In his dramatic works you recognise the author of "The Seasons," but his works of that class lack the concentrated thought and rapid action essential to perfection. His poem on "Liberty" has some fine passages, yet, as a whole, must be pronounced heavy. More attractive is "The Castle of Indolence." "The Seasons" have been often printed apart from the rest. Thomson's collected works appeared first in 1730. See also "Essays on the Life and Writings of Saltoun and the poet Thomson, by the Earl of Buchan," London 1792; Aikin's "Essay on the Plan and Character of Thomson's Seasons;" Lives of the poet by Johnson, Anderson, Nicola, etc. We present as specimens of this poet passages from A. PASBOYRIC OX OEEAT BRITA.IK. Heavens! what a goodly prospect spreads around, Rich is thy soil, and merciful thy clime; Full are thy cities with the sons of art; Bold, firm, and graceful, are thy generous youth; Thy sons of glory many! Alfred thine, Withstood a brutal tyrant's lustful rage, Like Cato firm, like Aristides just, Like rigid Cincinnatus nobly poor, A dauntless soul erect, who smiled on death. Frugal and wise, a Walsingham L- thine; A Drake who made thee mistress of the deep, And bore thy name in thunder round the world. Then flamed thy spirit high; but who can speak The numerous worthies of the maiden reign? In Raleigh mark their every glory mix'd, Raleigh, the scourge of Spain; whose breast with all The sage, the patriot, and the hero burn'd. May my song soften, as thy daughters I, Island of bliss! amid the subject-seas, 0 Thou! by whose almighty nod the scale the ratio of a : h is equal to that which is compounded of the ratios of a : b, of b : c, of c : d, of d : h. For the compound . , . abed a , ratio by the last article is -—- = — or a : h. 'be dh h A particular class of compound ratios is produced by multilying a simple ratio in itself, or into another equal ratio. hese are termed duplicate, triplicate, quadruplicate, etc., according to the number of multiplications. A ratio compounded of two equal ratios, that is, the square of the simple ratio, is called a duplicate ratio. One compounded of three, that is, the cube of the simple ratio, is called triplicate, etc. In a similar manner the ratio of the square roots of two quantities, is called a subduplicate ratio; that of the cubs roots a vbtriplieate ratio, etc. Thus the simple ratio of a to 4 is a : b The duplicate ratio of o to b is a5: b1 The triplicate ratio of a to 4 is o3 : 6* The subduplicate ratio of a to ft is •«/ •: The subtriplicate ratio of a to b is 3y' a i 3i/ 4, etc. N.B. The terms duplicate, triplicate, etc., must not be confounded with double, triple, etc. The ratio of 6 to 2 is 6:2 = 3 Double this ratio, that is, twice the ratio, is 12 : 2 = 6 Triple the ratio, i. e. three times the ratio, is 18 : 2 = 9 The duplicate ratio, i.e. the square of the ratio, is 6': 2s = 9 The triplicate ratio, I.«. the cube of the ratio, is 8*: 2* = 27 That quantities may have a ratio to each other, it is necessary that they should be so far of the same nature, that one can properly be said to be either equal to, or greater, or less than the other. Thus a foot has a ratio to on inch, for one is twelve times as great as ihe other. From the moae of expressing geometrical ratios in the form of a fraction, it is obvious that the ratio of two quantities is the same as the value of a fraction whose numerator and denominator are equal to the antecedent and consequent of the given ratio. Hence, To multiply or divide both the antecedent and consequent by the same quantity, does not alter the ratio. To multiply or divide the antecedent alone by any quantity, multiplies or divides the ratio; to multiply the consequent alone, divides the ratio; and to divide the consequent, multiplies the ratio. That is, multiplying and dividing the antecedent or consequent has the same effect on the ratio, as a similar operation, performed on the numerator or denominator, has upon the value of a traction. If to or from the terms of any couplet, two other quantities having the same ratio be added or subtracted, tlte sums or remainders will also have ihe same ratio. Thus the ratio of 12 : 3 is the same that of 20 : 5. And the ratio of the sum of the antecedents 12 + 20 to the sum of the consequents 3 + 5, is the same as the ratio of either couplet. That is, 12 + 20 12 20 l=J = i The ratio compounded of these is 72 : 12=6 Here the compound ratio is obtained by multiplying together the two antecedents, and also the two consequents of the simple ratios. Hence it is equal to the product of the simple ratios. Compound ratio is not different in its nature from any other ratio. The term is used to denote the origin of the ratio in particular cases. If in a series of ratios the consequent of each preceding couplet is the antecedent of the following one, the ratio of the first antecedent to the last consequent is equal to that which is com 'I of all the intervening ratios. Thus, in the series of ratios 2. Which is the greater, the ratio of a + 3 : 3. If the antecedent of a couplet be 65, and the ratio 13, what is the consequent i 4. If the consequent of a couplet be 7, and the ratio 18, What is the antecedent? 5. What is the ratio compounded of the ratios of 3 : 7, and 2a : Si, and 7*+ 1 : 3y —2? 6. What is the ratio compounded of * + y : b, and x — y: a + *, and a + b : A? 7. If the ratios of Bx + 7 : ix— 3, and * + 2 : \x + 3 be compounded, will they produce a ratio of greater inequality, or of less inequality? 8. What is the ratio compounded of x-\-y : a, and* — y : S, s> —v1 and *: y-? a 9. What is the ratio compounded of 7 : 6, and the duplicate ratio of 4 : 9, and the triplicate ratio of 3 : 2? 10. What is the ratio compounded of 3 : 7, and the triplicate ratio of x : y, and the subduplicate ratio of 49 : 9 i Proportion. When four quantities are related to one another in such a manner that the first divided by the second is equal to the third divided by the fourth—in other words, when the ratio of the first to the second is equal to the ratio of the third to the fourth, the four are said to be in direct proportion. From this definition it will be Been, that proportion is simply the equality of ratios. Though we have only spoken of two equal ratios, there may be any number, and in all cases the terms of these ratios are said to be in direct proportion. Care must be taken not to confound proportion with ratio. This caution is the more necessary, as in common discourse the two terms are used indiscriminately, or rather, proportion is used for both. The expenses of one man are said to bear a greater proportion to his income than those of another. But according to the definition which has just been given, one proportion is neither greater nor less than another. For equality does not admit of degrees. One ratio may be greater or less than another. The ratio of 12:2 is greater than that of 6:2, and less than that of 20: 2. But these differences are not applicable to proportion, when the term is used in its technical sense. The loose signification which is so frequently attached to this word, may be proper enough in familiar language; for it is sanctioned by general usage. But for scientific purposes, the distinction between proportion and ratio should be clearly drawn, and cautiously observed. Proportion may be expressed, either by the common sign of equality, or by four points between the two couplets. Thus \ 8 " 6 = 4 " 2« or 8 " 6 :: 4 " 2 i are arithmetical \ a — b — e — d, or a •• b :: e •• d\ proportions. And J 12 : 6 = 8 : 4, or 12 : 6 : : 8 : 4 ) are geometrical J o : b = d : A, or a : b : : d : A j proportions. The latter is read, "the ratio of a to b equals the ratio of The first and last terms are called the extremes, and the other two the means. Homologous terms are either the two antecedents or the two consequents. Analogous terms are the antecedent and consequent of the same couplet. As the ratios are equal, it is manifestly'immaterial which of the two couplets is placed first. If a : b :: c : d, then c i i:: a : b. For if ~=^r then —=-?-. b d do The number of terms in a proportion must be at least four. For the equality is between the ratios of two couplets; and each couplet must have an antecedent and a consequent. There may be a proportion, however, between n three quantities. For one of the quantities may be repeated, so as to form two terms. In this case the quantity repeated is called the middle term, or a mean proportional between the two other quantities, especially if the proportion is geometrical. Thus the numbers 8, 4, 2, are proportional. That in, 8 ; 4 : : 4 : 2. Here 4 is both the consequent in the first couplet, and the antecedent in the last. It is therefore a mean proportional between 8 and 2. The last term is called a third proportional to the two other quantities. Thus 2 is a third proportional to 8 and 4. Inverse or reciprocal proportion is an equality between a direct ratio and a reciprocal ratio. Thus 4:2: : { : I; that is, 4 is to 2 reciprocally, as 3 to 6. Sometimes, also, the order of the terms in one of the couplets is inverted, without writing them in the form of a fraction. Thus 4 : 2 : : 3 : 6 inversely. In this case, the first term is to the second, as the fourth to the third; that is, the first divided by the second, is equal to the fourth divided by the third. When there is a series of quantities, such that the ratios of the first to the second, of the second to the third, of the third to the fourth, etc., are all equal; the quantities are said to be in continued proportion. The consequent of each preceding ratio is then the antecedent of the following one. N.B. Continued proportion is also In the preceding articles of this section, the general properties of ratio and proportion have been defined and illustrated. It now remains to consider the principles which are peculiar to each kind of proportion, and attend to their practical application in the solution of problems. Arithmetical Proportion And Progression. If four quantities are in arithmetical proportion, the sum cf tin extremes is equal to the sum of the means. Thus if a ■■ b:: h •• m, then a+«=i+h For by supposition, a—L—h—m And transposing — b and — m, a+»=4+A. So in the proportion, 12 •• 10 :: 11 -9, we have 12+9=10+11. Again, if three quantities are in arithmetical proportion, the sum of the extremes is equal to double the mean. If a -b : :b ■■ c, then, a—fc=i—c And transposing — b and—c, a+c=2*. Quantities, which increase by a common difference, as 2, 4, 6, 8, 10, etc., or decrease by a common difference, as 16, 12, 9, G, 3, etc., are in continued arithmetical proportion. Such a series is also called an arithmetical progression; and sometimes progression by difference, or cquidifferent series. When the quantities increase, they form what is called an ascending series, as 3, 5, 7, 9, 11, etc. When they decrease, they form a descending series, as 11, 9, 7, 6, 3, etc. The natural numbers, 1, 2, 3, 4, 5, 6, etc., are in arithmetical progression ascending. From the definition it is evident that in an ascending series, each succeeding term is found by adding the common difference to the preceding term. If the first term is 3, and the common difference 2; The series is 3, 5, 7, 9, 11, 13, etc. If the first term is a, and the common difference d , Then a + a"is the second term, a+a'+e'=3 + 2«" the third, a+2a+a=a+3a- the 4th, a+3a+<£a«+4<Z the 5th, etc. 1st 2nd 3rd 4th 5th And the series is a, o+a", a+2o*, a+3a", a+4a", etc. If the first term and the common difference are the same, the series becomes more simple. Thus if a is the first term, and also the common difference, and n the number of terms, Then a+o=2a, is the second term, 2a+a=3o, the third, etc. And the series is a, la, 3a, 4a, na. In a descending series, each succeeding term is found by subtracting the common difference from the preceding term. If a is the first term, and d the common difference, the 1st 2nd 3nd 4th 5th series is a, a — d, a — 2d, a — 3a", a — 4a", etc. In this manner we may obtain any term by continued addition or subtraction. But in a long series, this process would become tedious. There is a method much more expeditious. By attending to the series, 1st 2nd 3rd 4th Sth it will be seen that the number of times d is added to a, is one Um than the number of the term. Thus, The second term in a + d, i.e. a added to once d; The third is a -j- 2d, a added to twice d; The fourth is a + 3<i- "added to thrice d; etc. So if the series be continued, The 50th term will be a + 49a\ The 100th term a + 9y</. If the series be descending, the 100th term will be a — 99a\ In the last term, the number of times d is added to a, is one leu than the number of all the terms. If then d zzz the common difference, a — the first term, z = the last, n = the number of terms, we shall have in all cases, : = a + (n— 1) y. d; that is, 1. To find the last term of an ascending series. Add to the Jirtt term the product of the common difference into the number of terms minus one, and the sum will be 2. To find the last term of a descending series. IS om the first term subtract the product of the common difference into the number of terms minus one, and the remainder will be the last term. N.B. Any other term may be found in the same way. For the series may be made to stop at any term, and that may be considered, for the time, as the last. Thus the with term = a + (m — 1) X. d. Frob. 1. If the first term of an ascending series is 7, the common difference 3, and the number of terms 9, what is the last term f Ans. s = a + (n— 1) d = 7 (0 — 1) X 3 = 31. Prob. 2. If the first term of a descending series is 60, the common difference 6, and the number of terms 12, what is the last term? Ans. s = a — (rt — 1) d — 60 — (12—1) x 5 —- 5, Prob. 3. If the first term of an ascending series be 9, and the common difference 4, what will the Sth term be? Ana. f = a + (m — 1) X d — 9 + (5 — 1) X 4 = 25. There is one other inquiry to be made concerning a series in arithmetical progression. It is often necessary to find the sum of all Let us take, for instance, the series 3, 5, 7, 9, 11, A nd also the same inverted, 11, 9, 7, 5, 3, The sums of the terms will be, 14, 14, 14, 14, 14. Take also the series a a -\- d, a 4- 2d, a -\- 3d, a -\- 4d, And the same inverted, a -(- -trf, a -\- 'id, a -j- 2d, a -j- d, a The sums will be 2a+4a", 2a+id, 2a+4d, 2a+4tf", 2a-f id Hence it will be perceived that the sum of all the terms in the double series, is equal to the sum of the extremes repeated as many times as there are terms. Thus, The sum of 14, 14, 14, 14 and 14 = 14 X 5, And the sum of the terms in the other double series is (2a + id) X 5 But this is twice the sum of the terms in the single series. If then we put a = the first term, n r= the number of terms, z — the last, s - the sum of the terms, we shall have this equation, $ =z - ~^ " x n. Hence, To find the sum of all the terms in an arithmetical progression. Multiply half the sum of the extremes into the number of terms, end the product will be the sum of the given series. The two formula:, t = a + (n — l)a", and I—' ' ~X",COB tains five different quantities; viz., a, the first term; d, the common difference; n, the number of terms; z, the last term; and s, the sum of all the terms. From these two formula: others may be deduced by which, if any three of the five quantities are given, the remaining two may easily be found. The most useful of these formulas are the following:— By the first formula, 1. 77x last term, IB a + (n — 1) d j in which a, n and d arc given. Transposing (n — 1) d, 2. The first term, a=« + (n— 1) d; z, n and d being given. Transposing a in the 1st, and dividing by n — 1, 3. The common difference, d=- a, z and n being given. n — 1 Transposing and dividing, z — a 4. The number of terms, n = —-— |