cular would be as much entitled to the name altitude or height of the triangle to the base AB, as the perpendicular AH is to The the name altitude or height of the triangle to the base B C. same may be said of a perpendicular drawn from the ver

tex B.

B

DEFINITION 2.-The altitude of a parallelogram is the perpendicular which measures the distance Fig. 3. of its two parallel sides. Thus, in the A parallelogram A B D C (fig. 3), if a perpendicular be drawn from the point B, or from any other point in the side A B, to the opposite side CD, it will measure the distance of the parallel sides AB, CD, and will be the altitude of the parallelogram A B D C, to the base c D. If a perpendicular were drawn from the point c, or from any other point in the ide AC, to the opposite side B D, it would measure the distance f the parallel sides A C, BD, and would be the altitude of the parallelogram

ABCD, to the base B D.

DEFINITION 3.-The altitude of a trapezoid is the perpendicular which measures the distance of its two parallel sides. Thus, Fig. 4. in the trapezoid ABCD (fig. 4), if a perpendicular be drawn from the point D, or from any other point in the side a D to the opposite side B c, it will measure the distance of the parallel sides A D, B C, and will be the altitude of the trapezoid to

o

the base B C.

Fig. 5.

C

or

DEFINITION 4.-The altitude of any rectilineal figure, polygon, is the greatest of all the perpendiculars which can be drawn to any side, or to any side produced, assumed as the base, from the vertices of its different angles. Thus, in the polygon ABCDE (fig. 5), the perpendicular drawn from the vertex c to the base A E, being greater than the perpendiculars drawn from the vertices B and D, to the same base, is the perpendicular of the polygon ABCDE to the base A E.

с

A

D

F

DEFINITION 5.-The extent of surface contained within the boundary of any plane figure, is called its area. Thus in the triangle A B C (figs. 1 and 2), the extent of surface contained within the three sides A B, BC, CA, is called the area of the triangle ABC. Again, in the parallelogram A CD B, the extent of surface contained within the four sides AB, BD, DC, CA, is called the area of the parallelogram A C D B.

DEFINITION 6.-Surfaces of ordinary extent are measured by the number of square inches, square feet, or square yards, which they contain, according as inches, feet, or yards, in length, have been used in taking their dimensions,-namely, their length, and their breadth. A square inch is a square whose side is one inch long. A square foot is a square whose side is one foot long. A square yard, is a square whose side is one yard long.

PROBLEM 1.-To find the area of a given square.-In order to determine the number of square inches, which are contained in a square of any size, as A B C D (fig. 6), measure the number of inches long in its side, and multiply this number by itself; the product will be the area or number of square inches which it contains. Thus, if A B the side of the square be measured, and found to be 6 inches long; then multiplying 6 by 6, you have 36 for the number of square inches in the square ABCD.

very obvious.

The reason of this process For, if the inches be carefully marked along the sides A B, B C, and straight lines parallel to these sides be drawn through each inchmark, it is plain that there will be 6 rows of six square inches; and these will all together contain 36 square inches; for 6X6

Fig. 6.

D

36. If the side of a square were measured and found to be 6 feet long; then, on the same principles, its area would be 36 square feet. Or, if the side of a square were measured and found to be 6 yards long; its area would be 36 square yards. In like manner, if the side of a square were measured and found to be 6 miles long, then, its area would be 36 square miles; and so on, for any other measurement, in inches, feet, yards, or miles. PROBLEM 2.—To find the area of a given rectangle. In order to

determine the number of square inches which are contained in Fig. 7. a rectangle of any size, as A B CD (fig. 7), measure the number of inches in its length, and the number of inches in its breadth or altitude; then multiply these two numbers together; and the product will be the number of square inches it contains. Thus, if the length AD measures 10 inches, and the breadth or altitude A B 4 inches, then, multiplying 10 by 4, you have 40 for the number of square inches in the rectangle ABCD. The reason of this process is also very obvious. For if the inches be carefully marked along the sides A D, A B, and straight lines parallel to these sides be drawn through each inch-mark as before, it is plain that, like the case of the square, there will be 10 rows of 4 square inches, and these altogether will contain 40 square inches; for 4x10 40. Also, as in the case of the square, the area of the rectangle will be in square inches, square feet, square yards, or square miles, according as the measurements of the dimensions (that is, of the length and breadth) are taken in inches, feet, yards, or miles of length.

PROBLEM 3.-To find the area of a given parallelogram. In order to determine the area of a parallelogram, draw a perpendicular from any point in the base to the opposite side, or to that side produced; measure the length of the base, and the length of the perpendicular or altitude, multiply these two lengths together, and their product will be the area of the parallelogram. This rule is founded on the 35th proposition of Book I., Euclid's Elements, in which it is demonstrated that parallelograms upon the same base and between the same parallels are equal. Hence, if a rectangle and a parallelogram stand on the same base and between the same parallels, they are also equal. Now the breadth of the rectangle is the same as the breadth of the parallelogram; whence the area of the parallelogram is obtained, by finding the area of the rectangle that stands on the same base, and has the same breadth or altitude,namely, the distance between the parallels. It is plain that the length of the oblique side of the parallelogram, that is, oblique to the assumed base, is not to be taken into consideration, in calculating its area; for, if it extended to an indefinite number of miles between the parallels, still its area would be the

same.

PROBLEM 4.-To find the area of a given triangle. For this purpose draw a perpendicular from the vertex of any angle, to the opposite side considered as the base, or to the base produced, if necessary; measure the length of the base, and the length of the perpendicular or altitude, multiply these two lengths together, and half their product will be the area of the triangle. This rule is founded on the 41st proposition of Book I., Euclid's Elements, in which it is demonstrated that if a parallelogram and a triangle be on the same base and between the same parallels, the parallelogram is double of the triangle. Now, since the rectangle formed by the base and the perpendicular base and of the same breadth or altitude, it follows that the breadth or altitude, is equal to the parallelogram on the same area of this rectangle is double the area of the triangle. Hence, the truth of the rule is evident.

PROBLEM 5.-To find the area of a given trapezoid. For this purpose, draw a perpendicular from any point in the base to length of the base, the length of the opposite side, and the the opposite side, or to that side produced; measure the length of the perpendicular or altitude; then add the lengths of the two parallel sides, and multiply their sum by the length of the perpendicular, and half this product will be the area of the trapezoid. This rule is plainly founded on the principle that if a diagonal be drawn joining two opposite extremities of the parallel sides, it will divide the trapezoid into two triangles, whose areas might be found separately, by Prob. 4, or conjointly by this rule.

PROBLEM 6.-To find the area of a given rectilineal figure. Divide the given rectilineal figure into triangles by drawing straight lines from one angular point to another, as shown in fig. 8. Then find the area of each triangle in this figure by Prob. 4. Add all these areas together, and their sum will be the area of the figure. It will shorten the process a little, and save fractions in the operation, first, to multiply the lengths of the bases and perpendiculars of the triangles together; then,

to add all these products together, and take half their sum, for the area of the figure.

In fig. 8, the contour or boundary of the rectilineal figure is denoted by the full lines; and the straight lines drawn within it, to form the triangles necessary for determining its area, are denoted by the dotted lines. The perpendiculars requisite to be drawn and measured for the computation of the triangles, are not shown; but from the explanations already given, they can very easily be conceived,

Fig. 8,

A more convenient mode of finding the area of a rectilineal figure, and one more frequently used in practice, is to divide it Fig. 9. into trapezoids and triangles, as shown in fig. 9. Then find the area of each trapezoid by Prob. 5, and of each triangle by Prob. 4; add all these areas together, and their sum will be the area of the figure. In planning this division of the figure it is best to draw as near as possible a straight line across the middle of it, and then from the angular points or vertices of the angles on each side of this straight line to draw perpendiculars to it. In fig. 9 these perpendiculars are shown by the dotted lines. The contour or boundary, and the central line, are denoted by the full lines. It will be seen from the consideration of this figure, that the areas of most of the triangles will in this case require to be subtracted from those of the trapezoids of which they form a part, because they are on the outside of the figure. Cases, however, can easily be supposed in which they will have to be added,-viz., when they are within the figure. In all such cases the computer of areas must use his judgment, or he may produce a serious error in the result.

The problems and rules we have just given, are sufficient to enable the intelligent and careful student to measure the area of any rectilineal figure, polygon, or surface, that may be presented to him. The fact is that they are the foundation of all the common processes of mensuration and land-surveying. With a foot rule, or a yard measure, the student may, if he understands these problems, proceed to measure surfaces of all kinds bounded by straight lines, in engineering, carpentry, plastering, roofing, building, painting, &c. With a measuring chain and measuring rods, he may also proceed to measure fields, commons, estates, and even townships, that are tolerably level and accessible to the taking of measurements. When greater accuracy is required, he will learn from future lessons what is necessary to be done, to accomplish this end,

LESSONS IN ENGLISH.-No. II,

By JOHN R. Beard, D.D.
INTRODUCTORY.

LANGUAGE is the expression of thought by means of articulate sounds, as painting is the expression of thought by means of form and colour. The relations which subsist between our thoughts, when carefully analysed and set forth sytematically, give rise to logic. The laws and conditions under which the expression of our thoughts takes place form the basis of grammar. The logician has to do with states of the intellect, the grammarian is concerned with verbal utterances.

That there are laws of speech a cursory attention to the subject will suffice to prove. There is, indeed, no province of the universe of things but is subject to law. Each object has its own mode of existence, which, in conjunction with the sphere of circumstances in the midst of which it is, gives rise to the laws and conditions by which it is controlled. Accordingly language takes its laws from the organs by which sound is made articulate, from the culture of the intelligent beings by whom these organs are employed, from the purposes for which speech is designed, and from even the medium and other outward influences in union with which these purposes are pursued.

Were there no such laws the science of grammar could not exist The sciences are in each case a systematic statement of generalised facts, in other words of definite laws; and grammar rests on phenomena clearly ascertained, invariable in themselves, capable of being distinctly stated, and equally capable of being wrought into a system of general truths. In many instances, indeed, the facts with which grammarians have to deal present themselves in the actual state of language, in a fragmentary and almost evanescent condition. The quick and piercing eye, however, of modern philology has succeeded in detecting no few of these, and the highly-cultivated powers which have been applied to the subject, have been able of themselves to supply deficiencies, and to construct edifices out of ruins, Still many things remain involved in darkness; in relation to others sagacious conjecture has authorised only bare probability. These, however, are not embraced within the science of grammar. When doubt begins science ends. What is still unascertained or subject to difficulties remains to be explored, and can take its place as part of scientific grammar only when it has ceased to be a subject of doubt and debate.

If the conditions under which thought became speech had been in all cases the same, there would only have been one language on the face of the earth. Descending as mankind did from a common progenitor the various tribes would have spoken a common tongue. But diversities soon arose. The organs of speech, while in all cases they remain substantially the same, vary in minor particulars with each individual. Outward influences are most diversified. Men's pursuits were different almost from the first. Climate and soil change with every change of locality. And both original endow-. ments and the degree of culture superinduced by external influences (or what may be termed indirect education) would be as diverse as the tribes, not to say the individuals of which the species consisted. All these diversified influences would speedily beget varieties in speech which time would increase and harden into different languages.

From this diversity, there arise two kinds of grammar, the universal, the particular, Universal grammar is formed by studying language in general, by passing in review the several languages which exist (or most of them), and selecting and classifying those facts' which are common to all. Particular grammar is the result of the study of any one given language. By a careful consideration of the usages of the best English writers we discover what constitutes English grammar. If, after we have ascertained the laws of a number of separate languages, we then compare our discoveries one with another, and mark and systematise what we find common to them all, we compose a treatise on general grammar. Particular grammar resembles the anatomy of the human frame, and limits its teachings to one set of objects. Universal grammar is like comparative anatomy which treats of the general laws of animal life, as deduced from a minute study of the animal kingdom in general.

It is with particular grammar that I am here concerned ;-of the grammar of our nation, namely, the English, I have to treat,

Grammar and logic, or the laws of expression and the laws of thought are, we have seen, closely connected together in the nature of things. Not easily, then, can they be sundered in manuals of instruction. If separate they are related sciences; as being related to each other, they may afford mutual light and aid, Requiring separate treatment, they each give and receive illustration. Grammar assists the logician to put his thoughts into a lucid form; and logic assists the grammarian to make his utterances correspond to the exact analogy of his thoughts. No one can be a good grammarian who is without skill in logic; and no logician who neglects grammar can successfully convey his ideas to others.

But in a manual which proposes to handle the subject of grammar, and of English grammar, reference to logic must be tacit and latent; it may be felt, it must not be displayed. Yet, in at least one or two terms will our obligations to logic be more positive and outward, for I shall borrow from that science, the words subject, attribute, predicate, &c.; and this I shall do, because these terms, when once their import is understood, afford facilities for explanation far greater than the ordinary terms employed in English grammars. In these cases, however, and in other things in which I shall depart from what is usual, I shall also supply the customary views and the ordinary terms.

As the English language, like other languages, was spoken before its laws were formed into a systematic treatise called a grammar, so the real facts of the language in their primary and

their model form, exist and are to be looked for in the every-day speech of well-educated mothers and fathers. But for the constant change, to which language is subject, I should not have needed to add the qualifying epithet "well-educated." But as language changes, so grammar changes; and thus what was good grammar under the Tudors is not good grammar in the age of Queen Victoria. Consequently it is not all usage that is of authority, but only the usage of the educated. Yet what is now educated usage will by and by become bad grammar and be accounted vulgar. So has it been in the past. Many of the present inaccuracies of the uneducated once possessed the authority which belongs to cultured speech, Provincialisms in word, in idiom, and in pronunciation may be traced back to lips which of old gave laws to other fashions besides the fashion of utterance. It may seem strange, but strange as it seems it is true, that among our forefathers legislators talked and harangued in terms and in tones the faithful representatives of which may now be heard at the ploughtail and in the smithy. Yet was that language the correct language of the day. And it was the correct language of the day because it was the language of the educated. Hence the speech of educated persons is of authority in grammar no less than the language of the best authors. Nay, we seem likely to find a language in its greater purity when we take it from the lips of educated persons generally than when we derive it from the somewhat artificial shapes which it assumes in the learned or the popular volume. If so "household words" are good for grammar as well as for practical wisdom. And so it is in the nursery we may look for the English tongue in a form the most simple and yet the most idiomatic. Of all teachers of English grammar the best is a well-educated English mother. Speaking from her own Saxon soul, and speaking to her own Saxon offspring, she pours forth from that "well of English undefiled," the Saxon element of our language, a stream of words and sentences which are sterling coin of the royal mint, current in all parts of the kingdom, the very substratum of English thought, whether found in books, in living speech, or in time-honoured institutions. Hence it is evident that a nursery in a cultivated English home, is the best school of English grammar. As a matter of fact, it is in such schools that, among the upper classes of this country, the young learn to speak correct English from their earliest days. Were all English children trained in such schools, the language would be everywhere well, and grammatically spoken. Consequently, could we place our students in cultivated nurseries, they would soon speak and write their mother tongue with correctness and propriety. We are unable to accomplish this. In nurseries of a different kind have they been brought up. They have been in schools, that is, their own houses, where they have learnt inaccuracies, where they have formed practices wrong in word, wrong in pronunciation, wrong in form. They have, therefore, not only to learn the right, but they have also to unlearn the wrong. A twofold difficulty attaches to their task. This twofold difficulty I shall constantly bear in mind, while I endeavour to introduce into these pages the language and the training of a cultivated English home, in such a way as to exhibit the fundamental usages and essential laws of the English language, as spoken by educated persons, and written by first-class authors. I cannot place the young of the working classes in cultivated nurseries, but I may attempt to do the next best thing; and that is, to bring forth and set before them in a living and organic form, the spoken language of such nurseries. And this shall I undertake, the rather because, as the mother is the child's natural educator, or, to speak more correctly, as the mother is an educator of God's own appointment, so every system of education will be good and effectual in proportion, as it is in form, substance, and spirit, motherly.

CORRESPONDENCE.

DOUBLE POSITION.

A correspondent requests an intelligible solution to the following question, No. 773, in "Walkingame's Arithmetic" (1826), under the rule of Double Position -

"A man had 2 silver cups cf unequal weight, having one cover to both of 5 oz. now if the cover be put on the less cup, it will be double the weight of the greater; and if set on the greater cup, it will be thrice as heavy as the less; what is the weight of each?"

and the errors to be noted; the 1st supposition to be multiplied by the second error, and the 2nd supposition by the first error; and then the difference of the products divided by the difference of the errors, when the errors are both in excess or both in defect; but the sum of the products divided by the sum of the errors, if one be in excess and the other in defect. In this question suppose the less cup to weigh 7 ounces, then adding to this 5 ounces (the cover), the weight is 12 ounces; this is double the weight of the greater cup, which accordingly weighs 6 ounces. Now, adding 5 ounces to this, the weight of the greater cup and cover is 11 ounces; but by the question, this weight is three times 7 ounces, or 21 ounces, this gives an error of 10 ounces in excess,

Again, suppose the less cup to weigh 9 ounces; then, adding to this, 5 ounces (the cover), the weight is 14 ounces; this is double the weight of the greater cup, which accordingly weighs 7 ounces. Now, adding 5 but by the question, this weight is three times 9 ounces, or 27 ounces; ounces to this, the weight of the greater cup and cover is 12 ounces; this gives an error of 15 ounces in excess.

Now, the product of the first supposition 7, by the second error 15, is 105; and the product of the 2nd supposition 9, by the first error 10, is 90; whence, the difference of these errors is 15; but the difference of the errors is 5; whence, 15 divided by 5, gives 3 ounces for the weight of the less cup.

Now, to find the weight of the greater cup, add the weight of the cover, 5 ounces to 8 ounces the weight of the less cup, and the whole is 8 ounces; by the question, this is double the weight of the greater cup, which is accordingly 4 ounces. Thus, the answer is 3 ounces the less cup and 4 ounces the greater cup.

ounces, the weight of the greater cup, and you have 9 ounces; by the The proof is as follows:-Add 5 ounces, the weight of the cover, to 4 question, this is three times the weight of the less cup, which accordingly is 3 ounces, as it ought to be.

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