fields. Subsequent discoveries have added numbers of specimens to the list of carboniferous reptiles, as well as many insects of grasshopper and locust affinities. These have been beautifully preserved in the fine-grained clay ironstone nodules of Saarbrück, near Treves. Beetles and white ants have also been found. These insects are usually associated with the leaves of ferns, and other such plants, clinging to which they probably died, and were encrusted in the deposition then going on. Clay-ironstone is one of the valuable products of the coal measures. It seems that the iron which reddened the sandstone of the Devonian period appeared in larger quantity in this epoch, and collecting in nodules, like the flints of the cretaceous age, forms every now and then bands of a valuable deposit. These nodules collect round some nucleus, generally the shells of the modiola, just as the flints form themselves round the sponges. The main source of the English iron is this deposit in the coal measures. Large quantities of it are exhumed near Bradford, the celebrated Lowmoor iron being produced from it. It is an impure carbonate of metal. LESSONS IN ENGLISH.-—XLVI. ADJECTIVES (continued). Nor do those admit of comparison which signify the stuff or substance of which anything is made, as golden, silver, brazen; nor those which denote the country to which persons belong, as British, Irish, French. Equally without comparison are all adjectives which indicate a fixed or definite idea, quality, or quantity, as square, triangular, round, circular, straight, upright, perpendicular, daily, annual, biennial. Moreover, adjectives which denote the highest state or condition of a quality or thing are without comparison. Thus you cannot compare boundless; for if an object is boundless, it cannot be more or less boundless. Accordingly, all words in less are incapable of comparison. Such are the adjectives extreme, full, immense, infinite, perfect, principal, right, wrong, supreme, preferable, universal. Sometimes, however, a latitude (not to be imitated) is taken in some of these words. Thus "more perfect" is said when "less imperfect" is meant, or more nearly approaching perfection." Equally incorrect is “ more certain;" "less uncertain" is what is intended. One class of adjectives has connected with it the epithet 'Numeral," because it contains the numbers. When we say "five persons," we obviously in five make use of an adjective, for the word five qualifies the word persons. Numeral adjectives are either cardinal or ordinal: the cardinal, one, THE following adjectives are usually given as irregular: two, three, four, etc., are those which form the series of our namely― Pos. Bad Far Good Little Much Comp. worse farther better less or lesser more Sup. worst. farthest. best. least. most. Here some explanations are necessary. Worse properly is not the comparative of bad, but of an adjective in the positive condition which has long ceased to exist. So bad itself has no comparative and no superlative in the present state of the language. Worst is nearly a regular form from worse. The same remarks may be applied to good, better, best. The full form in each case is made up by the aid of two distinct words. Better is the comparative of the Anglo-Saxon bet, as worse, worst are from the Anglo-Saxon vyrs. These are instances not so much of irregular as defective comparisons. Little, less, least seem to be abbreviated forms of little, littler, littlest; littlest became least, and least gave rise to less. Lesser, a corruption of less, seems to have arisen from the general form of er as a comparative sign, combined with the disinclination to a monosyllabic adjective. Most is a contracted form of the regular superlative from more, as more, morest, most; and more itself is a regular though shortened form from much, in the Anglo-Saxon, manig. In far, farther, farthest (Anglo-Saxon feor, fyrre, fyrrest) the th, as Latham remarks, seems euphonic, that is, inserted for the sake of ease in pronunciation. Far is now more common as an adverb than as an adjective. The word most becomes a terminational suffix to denote the superlative degree in these words, namely, foremost, hindmost, inmost, lowermost, outermost, undermost, uppermost. Here we have compound adjectives in the superlative, consisting, for instance, of the superlative most and the preposition under. Compare under with outer and upper. Outer is an adjective from out, as "He brought me into the utter (or outer) court" (Ezek. xlvi. 21). Outer (whence utter) has its counterpart in inner; thus, out, outer (utter), outmost (utmost), in, inner, inmost. What, then, is the origin of the preposition under? May it not be on and nether, that is, on the lower side; on nether = under; whence the superlative undermost. So upper is a comparative from the old positive up; up, upper (as in "upper-leather" in a shoe compared with the sole or under-leather), upmost. Most is often used without any direct reference to comparison, in order to denote a very high degree of the quality; as"Ere yet the salt of most unrighteous tears Had left the flushing in her galled eyes."-Shakespeare. The inflected superlative is used in a similar sense :---"Unbidden guests are often welcomest."-Shakespeare. Adjectives compounded with a noun defining their signification do not admit of comparison; as, blood-warm, coal-black, milkwhite, pea-green, sky-blue. numbers, being called "cardinal," because they are the chief ; thus, prudence, justice, temperance, and fortitude are called the cardinal virtues; that is, leading or most important, the principal virtues, as being those on the exercise of which the exist ence of others depends. The ordinal (Latin, ordo, rank, order), such as the first, the second, the tenth, are those which denote the place in a series which the numbers severally hold. The terms twice, thrice, etc., are units of a series which consists of what is called Numeral Adverbs. The numeral adverbs once, twice, thrice, denote the number of times a thing is to be taken; thus, twice two (or two twos) make four. The list is very defective, for after thrice we say four times, 'five times, etc., as, nine times nine make eighty-one. From the ordinal numbers first, second, third, etc., are formed by the addition of ly, so many adverbs, as eighthly, ninthly. Instead of these a periphrasis may be used, as in the first place, in the sixth place. These adverbs denote the sequence of topics or heads in a discourse or speech. The ordinals stand either before or after their nouns. Thus we say, "The third book," but speaking of a series of monarchs we place the adjective after the noun, as George the Fourth. We say the first of January, but we cannot, while our present queen is alive, speak of Victoria the First, because until there is a Victoria the Second, we are not sure that any series has been begun. With days of the month we say either the twelfth (12th) of May, or May the twelfth; the former is the more usual in ordinary affairs. There is yet another series of numeral adjectives, the multiplicative, as single, double, treble, quadruple, quintuple, centuple. These are immediately from the Latin. Another series of similar import are of Teutonic origin, namely, twofold, threefold, fivefold, etc. The comparison hitherto made involving more than is implied in the positive may be called the comparison of superiority. But all comparison involves the idea of less as well as the idea of more. The comparison which involves the idea of less may be called the comparison of inferiority. For the comparison of inferiority there are no inflexional terminations; it is formed exclusively by less and least, asPos. Rich Sly Able Comp. less rich less sly less able Sup. least rich. least sly. least able. Some adjectives in their nature do not admit of comparison. Such are those which end in ish, a termination like some, which denotes a portion, ish denoting a small portion; as, It is cold; yes, it is coldish; that is, rather, or somewhat cold. PRONOUNS-SIMPLE PRONOUNS. By its derivation a pronoun (pro, for, and noun, Latin nomen) is a for-noun, that is, a word used in place of a noun. This description may be accepted in a general way, yet will it scarcely bear a close investigation. Ia "I love," I is said to be a pro noun, but in place of what noun does I stand? In the statement, My shoes are worn out, my is said to be a pronoun; give me the noun whose place it holds. If I is a for-noun, for what noun is it? Clearly the definition is not satisfactory. Pronouns have been designated personal nouns. The definition is not without reason. I, for instance, is a noun, for it is the PERSON 1 name of a being. It is more than a noun, for it has specific reference to a person, and that person the first person, consequently I may be termed a personal noun. The connection of pronouns with persons is implied in the denomination of I, thou, he, etc., as personal pronouns. Pronouns are not found in the earliest stages of language. Society must have made considerable progress ere the minuter relations indicated by pronouns would receive appropriate terms. What are called the reflective pronouns, viz., self, himself, etc., seem to have been the latest in origin. This is exemplified in the case of the language of children. Before a child can say, I want my hat, he has learned to say, John wants John's hat, the repetition of the noun performing the office of the noun and the possessive pronoun. If we regard pronouns in their origin, we may divide them into two great classes-the primitive and the derivative: thus I is a primitive pronoun, and my, which comes from me, is a derivative pronoun. If we regard pronouns in their import, we obtain (1) Personal pronouns, or such as more specifically mark the person, as I, thou, he; (2) Possessive pronouns, or such as, besides marking the person, signify possession, as my and mine. If we regard pronouns in their office, we find some to be, (1) Relative, that is, having a specific reference to some preceding object, as who; (2) Interrogative, that is, such as are employed in asking questions, as who, which, and what; (3) Demonstrative, or those which demonstrate, that is, show or point out a person or thing, as this and that. Other pronouns are called (4) Reflective, such as self, oneself, themselves, because they reflect, or turn back, refer, that is, to the subject of the sentence. These, however, may also be termed compound, inasmuch as they consist of two pronouns; as, themselves consists of them and selves. Others, again, are called (5) Indefinite, such as some, any, whosoever, seeing that their import is vague and undetermined. There The primitive or personal pronouns are inflected in person, number, and case. They are inflected in person. In the nature of things as well as in grammar there are three persons. is the first person, or the speaker, I; there is the second person, or the person spoken to, thou; there is the third person, or the person spoken of, he. More persons than these there cannot be, for the speaker, the person spoken to, and the person spoken of, exhaust all possible modes of address. These three persons are in the singular number, denoting one individual. They may become plural; thus, I passes into we, thou into you, and he into they. These six forms as thus given are subjects. Corresponding forms denote objects; thus: Here we have an instance of genuine case, namely, a change of form corresponding with a change of signification. I, therefore, may be termed the nominative case and me the objective case; and so on. Only one of the personal pronouns is inflected in gender in the singular number, namely, he, masculine; she, feminine; it, neuter: the plural they is the same in the masculine, feminine, and neuter. We have already learned that the possessive my comes from the personal pronoun me (I); my and its corresponding forms are used before nouns, as my hat, thy shoe, his stick, our house, your garden, their aunt. Instead of my, mine is used if the pronoun stands without a noun; as, whose pencil is this? Answer, mine. The relative pronoun who is also inflected in case; thus, nominative who, possessive whose, objective whom. In the same manner is the interrogative declined or inflected, as who, whose, whom. With these explanations the reader will advantageously study this tabular view, in which are put together the chief facts connected with personal and possessive pronouns ; also relative and interrogative pronouns :— PERSON 2 PERSON 3 RELATIVE DEMONSTRATIVE INTERROGATIVE On the pronouns contained in this table we offer a few additional remarks. Thou is properly the pronoun of the second person singular. It is now, however, in common use, superseded by you, except in an emphatic personal address, in prayer and supplication made to the Divine Being. Ye The second person plural of the personal pronoun was formerly ye in the nominative and you in the objective case. is now almost obsolete in ordinary conversation, and you is employed for both the subject and the object. That mine was the original form of which my is a contraction, may be inferred from the fact that mine still stands before a noun beginning with a vowel, as— "Mine eyes have seen thy salvation." and thine are still common when they stand alone, the forms Mine, however, is in modern writing scarcely so used. Mine being retained probably from a regard to euphony, as my and thy would sound weak as terminating syllables. Our and your had of old forms in n (ourn, yourn) corresponding with mine and thine. Ourn became our, and this was lengthened into ours, the Saxon genitive. If so, ours and yours should be written our's s, probably as the sign of possession, being borrowed from the and your's. In his, originally neuter as well as masculine, you find s referred to things; as— "The mutes are those consonants whose sounds cannot be protracted."-Murray. Which is, in truth, only another form of who, and at the time when King James's translation of the Bible was made, which was used of persons as well as things. Which is still used of persons in such interrogations as, "Which of you will attend the lady?" Instead of who and which, that is often used as a relative, though properly that is a demonstrative. The two kinds of pronouns are, however, intimately connected together, inasmuch as who defines and that demonstrates; asRELATIVE.-The man who loved is loved. DEMONSTRATIVE.-The man? what man? that man. Another form of the relative who and which is what. What is used in asking questions, as, "What did you say?" "I said that you are ill:" you thus see that what corresponds to that; in other words, that is explanatory of what. In this case that has also a relative force, as may be seen in the fact that in Latin the word that in such a phrase would be the relative quod, and in French the relative que. What, in some phrases, comprehends both the relative and the antecedent, being equivalent to that which, as "This is what I wanted." A relative pronoun implies an antecedent. The antecedent is the word which goes before the relative, and to which the relative refers; as, "The seal is set. Now welcome, thou dread power, Nameless, yet thus omnipotent, which here Walkest in the shadow of the midnight hour."-Byron. Power is the antecedent, and which the relative: the sense shows that the relative which has reference to the antecedent power. This and that, demonstrative pronouns, differ thus: this points out the person or thing nearer to the speaker, that the more remote; as EXERCISE 10. 1. What is the length of a degree at the earth's equator, supposing the circumference of the earth at that part is 24,900 miles ? How 2. The radius of a circular plot of ground is 28 feet. many degrees will be contained in a portion of the circumference which measures 29.3216 feet ? PROBLEM VII.-To find the length of an arc, when the chords of the whole and of half the arc respectively are given, or when "Bring me that book." "This book ?" "No, that book which stands any lines in Fig. 16 are given by which these can be ascertained on the farthest part of the shelf." by previous rules. Rule: Subtract the chord of the whole are from 8 times the chord of half the arc, and divide the remainder by 3. EXAMPLE.-The chord BE of the whole arc (see Fig. 16) is 20, and the radius AC is 14; what is the length of the arc? To find BC, the chord of half the arc-A B, the radius, and BD BE (B) being known-proceed by Euc. I. 47. 4+4=8. Fig. 16. .. diameter Note. In this example the correctness of the rule is proved, since the diameter the chord B E, and half of this radius = 4; i.e., the height of c CD. EXERCISE 8. 1. The chord of an arc is 18.8, and the height is 8; what is the radius ? 2. In levelling for a canal, a certain allowance must be made for the curvature of the earth, since the line of sight is absolutely horizontal, in taking the level. This allowance it is found necessary to make to the extent of 8 inches per mile. Hence, what is the diameter of the earth? PROBLEM V.-Given the height of the arc, or the versed sine CD (Fig. 16), and the chord of half the arc B C, to find the radius. Rule: Divide the square of the chord of half the arc by the height of the arc; the quotient will be the diameter, which halve for the radius. Therefore the height of the arc is the square of the chord of half the arc divided by the diameter; and the chord of half the arc is the square root of the diameter multiplied by the height of the arc. EXAMPLE. The height of an arc is 16 feet, and the chord of half the arc is 32 feet; what is the diameter of the arc ? 322 1024 16 = 64 = diameter. EXERCISE 9. 1. The height of an arc is equal to half the chord of the whole arc, and this is 25 feet. What is the chord of half the arc ?* 2. The circular arch of a bridge rises 12 feet above the water, whose level touches the spring of the arch. The radius of the arch is 100 feet. How far is it in a direct line from the spring of the arch to its crown? 3. The arch of a bridge forms a part of a circle. The river beneath rises to the spring of the arch, and is 80 feet wide; and a boat's mast, 16 feet high, can just pass clear of the arch when in mid-stream. With what radius was the arch struck ? PROBLEM VI.-The circumference of a circle being given, to find the length of an arc of it, the number of degrees, etc., it contains being known. This is a simple and obvious case of proportion. Hence the Rule: As 360° (the number of degrees in the whole circle) is to the number of degrees, etc., in the arc, so is the length of the whole circumference to the length required. EXAMPLE.-The circumference of a circle measures 31,416 yards; what is the length of an arc of that circle containing =7,854 yards. 90° ? 360° 90°:: 31,416 yards: 31,416 4 The equality of the two given elements in this example shows them to be the radii of a circle; hence the chord of half the arc, being the hypothenuse, can be found by Euc. I. 47. EXERCISE 11. 1. The chord of the whole are is 36 feet 9 inches, and the chord of half the arc is 23 feet 3 inches. What is the length of the arc ? 2. The span of a circular arch is 48 feet, and the length from the spring of the arch to its crown is 30 feet in a direct line. How many stones of 9 inches each compose the arch? There might follow several other problems having reference to the mensuration of lines, such as the finding of the length of the diameter of a circular zone-that is, of the circle of which it forms a part; of the component parts of an ellipse, some of the parts being given; the same of a parabola; but all these. although interesting, are not of such importance as those we sideration of the next part of our subject, namely, the mensurahave given, and we shall therefore at once proceed to the contion of surfaces-that is, of spaces entirely enclosed by lines. The necessity for considering the measurement of lines as introductory to that of superficies will be apparent from a consideration of the fact that the area of surfaces are necessarily fore, a knowledge of the one gives us at once the key to a dependent upon the lines which enclose them, and that, thereknowledge of the other. 2. A square table is 3 feet across either side; how many squares of 1 inch could be marked out upon it? 3. A deal board is 11 feet long and 11 inches wide, the ends being square. I want to cut it up into pieces 1 foot long and 1 inch wide; how many can I cut ? 4. The area of a square field is 1 acre; how long is each side in links and yards? 5. A rectangular space, intended for planting, is 300 yards long and 220 yards broad. If I cut a path across it the longest way, 4 feet wide, how much space will remain available for planting? 6. A street, 30 feet wide and 1 mile long, has to be paved at a cost of 48. per square yard; what will the total cost be? 7. The adjacent sides of an acute-angled parallelogram are 20 and 15 feet; at what angle must they incline so that the area of the parallelogram shall be 259.8 feet? 8. What would be the area of the above figure if the angle were 30° instead of 60° ? PROBLEM IX.-The diagonal of a square being given, to find its area. Rule: Square the diagonal and halve the product. Hence the side of a square is to the diagonal as the square root of half its square. EXAMPLE 1.-The diagonal of a square is 10; what is its area? 3. The area of a square is 1 acre, 1 rood; what is the length of the diagonal? PROBLEM X.-To find the area of a triangle, the base and E altitude being given. Let A B C (Fig. 19) be a triangle, and A D its altitude. Rule: Multiply the base by half the altitude; the product is the area. Fig. 19. The truth of this is evident from Euclid I. 41, in which it is proved that if a triangle and a parallelogram be upon the same base, and between the same parallels, the triangle is one-half the parallelogram; for the area of the parallelogram ABCE is BCX AD; and hence that of the triangle ABC is BC X 2 EXAMPLE. The base of a triangle is 20, and the altitude is 10; what is the area of the triangle? 20 X = 100, area of triangle. 10 2 EXERCISE 14. A2 1. The base of a triangle is 43, and the altitude 21; required the area. 2. The base of a triangle is 150 yards, and the altitude 120 yards; required the area in acres, roods, etc. 3. The hypothenuse of a right-angled 18 triangle is 68, and the base 24; what is the area ? 20 12 C 7 D Fig. 20. 4. The side of an equilateral triangle is 6; what is its area? 5. The three sides of a triangle are respectively 20, 21, and 29 poles; required its area in acres, roods, and poles. PROBLEM XI. To find the area of a trapezium.* Pule Divide the figure into two triangles, by drawing a diagonal; then compute the areas of the triangles separately, by previous rules, and add these areas together. A trapezium is a quadrilateral figure (four-sided) in which no two of its sides are parallel. The problem may be extended to finding the area of any quadrilateral figure, The correctness of the working may be proved by drawing the opposite diagonal, and repeating the computation. The two results will agree if the calculation is correct. EXAMPLE 1.-The lengths of the four sides of a trapezium ABCD (Fig. 20) are as follow:- A B = 20; BC= 12; CD=7; and DA = 18, and the diagonal B D is 18. What is its area? In the triangle A B D, the two sides AD and D B being equal, its area is double that of A E D. Then ED=/182 202 2 = √324 - 100 = √224 = 15, nearly. 1382-875 •5(c) 691-4375 Area of B C D√691.4375 26.3, nearly. Then area of trapezium = areas of ABD + BCD or 150 +263 176.3. Ans. = In actual measurements, the diagonal A c may be ascertained, and the areas of the two triangles A B C and ACD found. Their sum will be found to be as above, provided the measurements and calculations are correctly performed. Note. In the application of this, and any other rule for the measurement of surfaces as applied to land surveying, too many checks on the correctness of the results cannot be taken. EXERCISE 15. and 14; the diagonal across between the two most obtuse angles 1. The four sides of a trapezium are respectively 20, 16, 12, (draw the figure) is 14. Required the area of the trapezium. 2. The four sides of a trapezium are 628, 464, 457, and 733, and the diagonal from the angle between the two shortest sides the opposite angle is 835. Required its area. To this as well as to the other exercises that have been given as necessary appendages to the different problems, the learner can easily add examples for practice by substituting other numbers in the various examples in each exercise; or by drawing triangles, parallelograms, trapeziums, etc., according to scale, and working out their contents for their dimensions. |