the length taken for the unit, the product is a like part of the multiplicand. Thus, if one of the factors is 6 inches, and the other half an inch, the product is 3 inches. Instead of referring to the measures in common use, as inches, feet, etc., it is often convenient to fix upon one of the lines in a figure, as the unit with which to compare all the others. When there are a number of lines drawn within and about a circle, the radius is commonly taken for the unit. This is particularly the case in trigonometrical calculations. The observations which have been made concerning lines, "may be applied to surfaces and solids. There may be occasion to multiply the area of a figure by the number of inches in some given line. But here another difficulty presents itself. The product of two lines is often spoken of as being equal to a surface; and the product of a line and a surface, as equal to a solid. But if a line has no breadth, how can the multiplication, that ás, the repetition, of a line, producé a surface? And if a surface has no thickness, how can a repetition of it produce a solid? In answering these inquiries it must be admitted that measures of length do not belong to the same class of magnitudes with superficial or solid measures; and that none of the steps of a calculation can, properly speaking, transform the one into the other. But, though a line cannot become a surface or a solid, yet the several measuring units in common use are so adapted to each other, that squares, cubes, etc., are bounded by lines of the same name. Thus the side of a square inch is a linear inch; that of a square rod, a linear rod, etc. The length of a linear inch is, therefore, the same as the length or breadth of a square inch. If then several square inches are placed together, as from o to R, fig. 3, the number of them in the parallelogram o R is the Fig. 3. D lengths; and, if the length of each were the same, the areas A: 4 :: L: 7, when the breadth is given; The area of an oblique parallelogram is also obtained by Thus multiplying the base into the perpendicular height. the expression for the area of the parallelogram A BN M, fig. 5, is M N X AD, or A B X B C. For ABX BC is the area of the Fig. 5. Fig. 6. R right-angled parallelogram ABCD; and by Euclid 36, 1, For the area is equal to ABX BC. But AB BC, therefore, AB X B CAB X AB = (A B)a. The area of a triangle is equal to half the product of the base and height. Thus the area of the triangle AB G, fig. 7, is equal to half ▲ B into G H or its equal BC, that is, a = ABX BC. same as the number of linear inches in the side a R: and if We hence obtain a convenient algebraical expression for the area of a right-angled parallelogram. If two of the sides perpendicular to each other are A B and B C, the expression for the area is A B X BC; that is, putting a for the area, @ABX BC. It must be remarked, however, that when A B stands for a line, it contains only linear measuring units; but when it enters into the expression for the area, it is supposed to contain superficial units of the same name. The expression for the area may also be derived by another method more simple, but less satisfactory perhaps to some. Let a, fig. 4, represent a square inch, foot, rod, or other measuring unit: and let b and be two of its sides. Also, For the area of the parallelogram ABCD is ABX BC. And by Euclid 41, 1, if a parallelogram and a triangle are upon the same base, and between the same parallels, the triangle is haif the parallelogram. Hence, an algebraical expression may be obtained for the area of any figure whatever which is bounded by right lines. For every such figure may be divided into triangles. Thus the right-lined figure, A B CBE, fig. 8, is composed of the triangles A B C, A CE and E CD. The area of the whole figure is, therefore, equal to (ACX BL) + ( A ©× EH) + (} ECX DG). The expression for the superfices has here been derived from that of a line or lines. It is frequently necessary to reverse this order; to find a side of a figure, from knowing its area. If the number of square inches in the parallelogram A B C D, fig. 3, whose breadth B C is 3 inches, be divided by 3, the quotient will be a parallelogram, A B EF, one inch wide, and of the same length with the larger one. But the length of the small parallelogram is the length of its side A B. The number of square inches in one is the same as the number of linear inches in the other. If, therefore, the area of the large That is, the base of a triangle is found, by dividing the area by half the height. As a surface is expressed by the product of its length and breadth, the contents of a solid may be expressed by the product of its length, breadth and depth. It is necessary to bear in mind, that the measuring unit of solids is a cube; and that the side of a cubic inch is a square inch; the side of a cubic foot, a square foot, etc. Let ABCD fig. 3, represent the base of a parallelopiped, five inches long, three inches broad, and one inch deep. It is evident there must be as many cubic inches in the solid, as there are square inches in its base. And as the product of the lines ▲ B and B c gives the area of this base, it gives, of course, the contents of the solid. But suppose that the depth of the parallelopiped, instead of being one inch, is four inches. Its contents must be four times as great. If, then, the length be ▲ B, the breadth в c, and the depth co, the expression for the solid contents will be, ABX BCX CO. By means of algebraical notation, a geometrical demonstration may often be rendered much more simple and concise than in ordinary language. The proposition, (Euclid 4, 2,) that when a straight line is divided into two parts, the square of the whole line is equal to the squares of the two parts, together with twice the product of the parts, is demonstrated, by squaring a binomial. Let the side of a square be represented by s; And squaring both sides, s = a+b; $2 a2+2ab+b2. That is, s2 the square of the whole line, is equal to a and b, the squares the two parts, together with 2ab, twice the product of the parts. Algebraical notation may also be applied, with great advantage, to the solution of geometrical problems. In doing this, it will be necessary, in the first place, to form an algebraical equation from the geometrical relations of the quantities given and required; and then by the usual reductions, to find the value of the unknown quantity in this equation. Prob. 1. Given the base, and the sum of the hypothenuse and perpendicular, of the right-angled triangle A B C, fig. 9, to find the perpendicular. B In a right angled-triangle, the perpendicular is equal to the square of the sum of the hypothemuse and perpendicular, diminished by the square of the base, and divided by twice the sum of the hypothenuse and perpendicular. the letters a and b. Thus if the base is 8 feet, and the sum of 162-82 becomes =6, the perpendicular; and this subtracted 2 X 16 from 16, the sum of the hypothenuse and perpendicular, leaves 10, the length of the hypothenuse. Prob. 2. Given the base and the difference of the hypothenuse and perpendicular of a right-angled triangle, to find the perpendicular. Prob. 10. If the sum of two of the sides of a triangle be 1155, the length of a perpendicular drawn from the angle included between these to the third side be 300, and the differ ence of the segments made by the perpendicular be 495; what are the lengths of the three sides? Prob. 11. If the perimeter of a right-angled triangle be 720, and the perpendicular falling from the right angle on the hypothenuse be 144; what are the lengths of the sides? Prob. 12. The difference between the diagonal of a square and one of its sides being given, to find the length of the sides. Prob. 13. The base and perpendicular height of any plane triangle being given, to find the side of a square inscribed in the triangle, and standing on the base, in the same manner as the parallelogram D E F G, on the base AB, fig. 14. Prob. 14. Two sides of a triangle, and a line bisecting the included angle being given, to find the length of the base or third side, upon which the bisecting line falls. Prob. 15. If the hypothenuse of a right-angled triangle be 35, and the side of a square inscribed in it, in the same manner as the parallelogram BEDF, fig. 12, be 12; what are the lengths of the other two sides of the triangle? Ne paraissez jamais ni plus sage ni plus savant que ceux avec qui vous étes. Portre votre savoir comme votre montre, dans une poche particulière. que vous ne tirez point, et que vous ne faites point sonner uniquement pour nous faire voir que vous en avez une.-Lord Chesterfield. coming having come Past Participle: venúto, come INDICATIVE MOOD. Present. venimmo, we came Véngo or végno, I come vieni, thou comest viene, he comes veniamo or vegnámo, we come veníte, you come véngono or végnono, they come Imperfect. Veniva or venía, I was coming venivi, thou wast coming veniva or venis, he was coming venivámo, we were coming venivate, you were coming venivano, venieno, or veniano, they were coming Indeterminate Preterite. Vénni, I came venisti, thou camest vénne, he came veniste, you came vénnero or veníro, they came Verrò, I shall or will come Conditional Present. Verrei or verría, I should or would come verrésti, thou wouldst come verrébbe or verría, he would come verrémmo, we would come terreste, you would come verrebbero, they would come IMPERATIVE MOOD. [No First Person.] Vieni, come (thou) vénga, let him come Present. veniamo, let us come veníte, come (ye or you) véngano, let them come SUBJUNCTIVE MOOD. Imperfect. Che venissi or venéssé, that I might come che venissi, that thou mightst come che venisse, that he might come che venissimo, that we might come che veniste, that you might come che venissero, that they might come Present: pióvere, to rain rained let there be Past avere piovuto, to have Ci or vi sia, or staci or síavi, | Ci or vi siano, or sîanci or síanvi, let there be SUBJUNCTIVE MOOD. Present Gerund: piovendo, Past Gerund: avendo piovúto, raining Past Participle: piorúto, rained Present. Pure, it rains Imperfect. Pioveva, it rained having rained INDICATIVE Moon. Indeterminate Preterite. Piove, it rained Future. Piorerà, it will rain Conditional Present. The Participle is a word which possesses the qualities both of the verb and the adjective. Participles are of three kinds :-Present, Past and Future. 1. The Present Participle terminates in ando or endo; as Amando, loving 2. The Past Participle ends as follows in the regular verbs : Amato-a, amati-e, loved Creduto-a, crediti-e, believed Servito-a, serviti-e, served 3. The Participle Future is not so often used. It is as follows: Avére ad amare, éssere per amáre, being about to love Avére a credere, éssere per crédere, being about to believe Avére a servíre, éssere per servire, being about to serve The Italians are accustomed to syncopate several past participles of the first conjugation, as may be seen in the following list: fitted accustomed laden INDICATIVE MOOD. Future. Bisogna, it is necessary Bisognerà, it will be necessary Imperfect. Conditional Present. avvezzáto cérco compro cóncio casso Quánto prima, as soon as possible L'anno prossimo, next year Da qui a due mési, in two months time Il giorno seguente, the next day Domani mattina, domattina, to morrow morning D'óra innnázi, henceforward Of Time Indefinite. Alle volte, sometimes Italian adverbs are formed from adjectives in three ways, Di continuo, continually viz.: 1. By uniting the substantive mente to the feminine of the adjectives ending in a; as— Dótto or dótta, learned saggio or saggia, wise váno or vána, vain Dottamente, learnedly saggiamente, wisely vanamente, vainly Di di in di, di giórno in giórno, from day to day Di giorno, by day Di quando in quando, di tempo in témpo, trátto, tratto, from time to time | Fin dove, till where Sù in alto, di sópra, above or up stairs Fin a quándo, till when Di sótto, abbásso or giù, under D' intorno, all around Un dopo l'altro, alla fila, one after another Insieme, together 2. By adding the substantive mente to the adjectives ending Fin adésso, fin a quest óra, fin A uno a uno, one by one in e not preceded by 7; as Il mense passato, last month Fra poco, in breve, shortly A domani dunque, to-morrow All avvenire, in future óra, hitherto Fra póco, in a short time Il più sovente, oftener In breve, in breve tempo, shortly In quel mentre, in the mean time In tempo, seasonably A due a due, two by two In seguito, di seguito, after- Dálle fondamenta, from the foundation In un batter docchio, all of a Infine, álla fine, at last sudden In un attimo, all at once Per sempre, for ever Non ancóra, not yet Dóve, where Di lì, là, from thence Altrove, somewhere else Sopratutto, above all Alla rinfusa, sossópra, topsy turvy A vicenda, alternately Of Quantity and Number. Quanto, how much Quási, almost Molto, much Troppo, too much Un pochettino, but a little Niente affatto, not at all Non molto, not much Per metà, by half Davvantaggio, di vantaggio, di più, some more Un poco di più, a little more Un poco di méno, a little less Un po' troppo, a little too much Molte volte, several times |