1 of a, i.e. that a is to be resolved into three equal factors; Obs. From the manner of performing evolution it is for a3 Xa Xa3 = a. On the other hand, denotes the evident, that the plan of denoting roots by fractional indices, third power of the fourth root of c, or the fourth root of the is derived from the mode of expressing powers by integral third power. One expression is equivalent to the other. Instead of a", we may write a. indices. Ans. a3, or 3y a. 1. Required the cube root of a. Ans. a3. 5. Required the seventh root of 2d-x. The rule in the preceding article may be applied to The index of a power or root may be exchanged for any every case in evolution. But when the quantity whose root other index of the same value. Instead of a we may put a For in the latter of these expressions, a is supposed to be resolved into twice as many factors as in the former; and the numerator shows that twice as many of these factors are to be multiplied together. Hence the value is not altered. From the preceding article it will be easily seen, that a fractional index may be expressed in decimals. 7. Thus aal, or a0-5; that is, the square root is equal to the fifth power of the tenth root. 9. Express a in decimals. 8. Express a in decimals. 3 10. Express a in decimals. 11. Express a in decimals. 11 12. Express a in decimals. The process of resolving quantities into equal factors, is called Evolution. In subtraction, a quantity is resolved into two parts. In division, a quantity is resolved into two factors. In evolution, a quantity is resolved into equal factors. Evolution is the opposite of involution. The former is finding a power of a quantity, by multiplying it into itself. is finding a root, by resolving a quantity into equal factors. A quantity is resolved into any number of equal factors, by dividing its index into as many equal parts. The other From the foregoing principles we deduce the following GENERAL RULE FOR EVOLUTION. Divide the index of the quantity by the number expressing the root to be found Or, Place the radical sign belonging to the required root over given quantity. the If the quantities have co-efficients, the root of these must be extracted and placed before the radical sign or quantity. Thus, To find the square root of d', divide the index 4 by 2, i. e. d. So the cube root of d3, is = d2, is to be found, is composed of several factors, there will frequently be an advantage in taking the root of each of the factors separately. This is done upon the principle that the root of the product of several factors is equal to the product of their roots. Thus ab va x vb. For each member of the equation if raised to any power, will give the same result. When, therefore, a quantity consists of several factors, we find the root of the factors separately, and then multiply them may either extract the root of the whole together, or we may SIGNS.-(1.) An odd root of any quantity has the same sign as the quantity itself. (2.) An even root of a positive quantity is ambiguous. But an even root of a positive quantity may be either positive or negative. For the quantity may be produced from the one, as well as from the other. Thus the square root of a2 is +a, or — a. An even root of a positive quantity is, therefore, said to be ambiguous, and is marked with the sign. Thus the square root of 36 is 36. The 4th root of a is ±xa. The ambiguity does not exist, however, when from the nature of the case, or a previous multiplication, it is known whether the power has actually been produced from a positive or from a negative quantity. But no even root of a negative quantity can be found. The methods of extracting the roots of compound quantities need not be considered here. But there is one class of them, the squares of binomial and residual quantities, which it will be proper to attend to in this place. The square of a + b, for instance, is a2+2ab+b2, two terms of which, a2 and b2, are complete powers, and 2ab is twice the product of a into b, that is, the root of a2 into the root of b2. Whenever, therefore, we meet with a quantity of this description, we may know that its square root is a binomial; and this may be found by taking the root of the two terms which are complete powers, and connecting them by the sign. The other term disappears in the root. Thus, to find the square root of x2 + 2xy + y2, take the root of a2, and the root of y, and connect them by the sign+. The binomial root will then be x + y. In a residual quantity, the double product has the sign prefixed, instead of +. The square of ab, for instance, is a2 2ab+b. And to obtain the root of a quantity of this description, we have only to take the roots of the two complete powers, and connect them by the sign. Thus the square root of 2 2xy + y2, is x-y. Hence, To extract the square root of a binomial or residual. Take the roots of the two terms which are complete powers, and connect them by the sign which is prefixed to the other term. 1. To find the root of 2 + 2x + 1. The two terms which are complete powers, are 2 and 1. 2. Find the square root of x2 - 2x + 1. 3. Find the square root of a + a + · 4. Find the square root a2+ fa+ $. 5. Find the square root of a + ab +· out any alteration of its value. For "yanan = a. 2. Reduce 4 to the form of the cube root. 3. Reduce 3a to the form of the 4th root. 4. Reduce fab to the form of the square root. 5. Reduce 3 X (ax) to the form of the cube root. 6. Reduce a to the form of the cube root. N B. In cases of this kind, where a power is to be reduced to the form of the nth root, it must be raised to the nth power, not of the given letter, but of the power of the letter. Thus, in the 6th example, a is the cube, not of a, but of a2. 7. Reduce asb to the form of the square root. 8. Reduce am to the form of the nth root. CASE II. To reduce quantities which have different indices to others of the same value having a common index. Reduce the indices to a common denominator. also each quantity to the power expressed by the mumereduced index. (3.) Take the root denoted by the common denominator. 9. lteduce at and to a common index. 1st. The indices and reduced to a common denominator are and. 2nd. The quantities a and b raised to the powers expressed by the two numerators are a3 and 62. 3rd. The root denoted by the common denominator is the Yth. The answer, then, is (a) and (b)14. The two quantities are thus reduced to a common index, without any alteration in their values. For aal, which = (a3) 1a ̧ m CASE III. To reduce a quantity to one with a given index. Divide the index of the quantity by the given index, place the quotient over the quantity, and set the given index over the whole. This is merely resolving the original index into two factors. 17. Reduce a one with the index. This is the index to be placed over a, which then becomes a; and the given index set over this, makes it (aš) †, a' the answer. 18. Reduce a ale to others with the common index}. -} = 2 × 3 = 6, the first index. 2 ÷ ÷ 1 = 1× 3 =, the second index. Therefore (a) and (z3) are the quantities required. 19. Reduce 42 and 3 to others with the common index. 20. Reduce x and y to others with the common index §. 21. Reduce a and b to others with the common index. 22. Reduce e2 and d to others with the common index 1. 2 1 23. Reduce am and bm to others with the common index 24. Reduce a, ¿, and a to others with the common index. CASE IV. To reduce a radical quantity to its most simple terms; i. e. to remove a factor from under the radical sign. Resolve the quantity into two factors, one of which is an exict power of the same name with the root; find the root of this power, and prefix it to the other factor, with the radical sign between them. This rule is founded on the principle, that the root of the product of two factors is equal to the product of their roots. It will generally be best to resolve the radical quantity into such factors, that one of them shall be the greatest power which will divide the quantity without a remainder. N. B. If there is no exact power which will divide the quantity, the reduction cannot be made. CASE V. To introduce a co-efficient of a radical quantity under the radical sign. grieving Past Participle: grieved dolútosi, INDICATIVE MOOD. Present. Raise the co-efficient to a power of the same name as the radical Mi dólgo or dóglio, I grieve part, then place it as a factor under the radical sign. ti duoli, thou grievest ci dogliamo, we grieve Imperfect. si dólse, he grieved ci dolémmo, we grieved Mi dorrò, I shall or will grieve they ti dorra, thou wilt grieve si dorrà, he will grieve Mi doleva or doléa, I grieved ti dolevi, thou grievedst si doléva or dolea, he grieved ci dolevámo, we grieved vi doleváte, you grieved si dolévano, they grieved Indeterminate Preterite. Mi délsi, I grieved ti dolesti, thou grievedst ci dorrémo, we will grieve vi dorréte, you will grieve si dorrúnno, they will grieve Conditional Present. Mi dorréi or dorría, I should or would grieve ti dorrésti, thou wouldst grieve si dorrébbe, he would grieve ci dorrémmo, we would grieve vi dorréste, you would grieve si dorrébbero, they would grieve IMPERATIVE MOOD. [No first Person.] dólga, let him grieve Dubliti, grieve (thou) 4. Reduce a2 and as to others with the common index.si 5. Reduce 98 to its simplest form. 6. Reduce 243 to its simplest form. 7. Reduce 54 to its simplest form. 8. Reduce 780 to its simplest form. 9. Reduce 9/81 to its simplest form. 10. Reduce ax to its simplest form. 11. Reduce 1982 to its simplest form. 12. Reduce √3a2 to its simplest form. Present. dogliamoci, let us grieve dolétevi, grieve (ye or you) si dolgano, let them grieve SUBJUNCTIVE MOOD. So conjugate Condolérsi to complain. | Ridolére, to lament again. Observation.-Dolére signifies, also, to ache, and then it is used impersonally. Examples: Mi duble la tésta, my head aches, or I have the headache. Mi dolgono i denti, my teeth ache, or I have the toothache. V. Dovére, to owe. INFINITIVE MOOD. Simple Tenses. Present: dovere, to owe Compound Tenses, Past avére dovuto, to have owed Present Gerund: dovendo, Past Gerund: avendo dovuto, owing Past Participle: dovuto, owed having owed down down giace mmo, we lay down Future. giacerái, thou wilt lie down IMPERATIVE MOOD. [No First Person.] Giáci, lie (thou) down giacciamo or giacidmo, let us lie down giaccia or giácia, let him lie | giace'te, lie (ye or you) down down Present. giacciano or giáciano, let them lie down SUBJUNCTIVE MOOD. Che giaccia or guúcia, that I may che giaccia, giácia, giacci, giáci, we may lie down you may lie down VII. Parere, to seem. Simple Tenses. Present Gerund: pare'ndo, Past Participle: párso or pa ruto, seemed Present. Pájo, † I seem Imperfect. INDICATIVE MOOD. pare'mmmo, we seemed Parrò, I shall or will seem parrái, thou wilt seem Iparra, he will seem parremo, we will seem parre'te, you will seem parránno, they will seem Conditional Present. Parre'i or parria, I should or would seem they parre'sti, thou wouldst seem parrebbe or parria, he would Indeterminate Preterite, Parvi or pársi, I seemed parve or párse, he seemed seem parre'mmo, we would seem parre'ste, you would seem parrebbero or parriano, they would seem • Paruto is not so often used, and not so good as párso. + Some Italians have used paro, pariámo, and párono, instead of pajo, • Some Italians think that there is no imperative; but as excellent gram- pajamo, and pájano. The learner must take care not to imitate them, for marians have given it, we prefer their opinion, the former come from paráre, to adorn. Verbs in 2. a-w Past Gerund: avendo per- 3. ε-w 4. ευ-ω μισθο-ω, I hire χρυσο-ω, 1 gild ζημιο-ω, Ιρακish δικαζω, I judge σημαιν-ω, I signify λευκαιν-ω, I whiten Noun whence derived. μiolo-, wages, reward. χρυσο-ς, gold. airi a, cause, blame. Þıdıñños, Philip. ἔργον, labour. Bia, force. σῆμα, a sign. Persuado, I persuade persuadi, thou persuadest persuade, he persuades persuadidmo, we persuade persuadé te, you persuade persuadono, they persuade Imperfect. Persuade'va, I persuaded persuade vi, thou persuadest persuade'va, he persuaded persuadevamo, we persuaded persuadeváte, you persuaded persuadé vano, they persuaded Indeterminate Preterite. Persuasi, I persuaded persuadesti, "thou persuadedst persuase, he persuaded persuade mmo, we persuaded persuade'ste, you persuaded persuasero, they persuaded persuaderái, thou wilt persuade suade persuaderánno, they will per- Conditional Present. thou wouldst persuadere'mmo, we would per- persuadere'ste, you would per- persuaderebbero, they would 8. υν-ω λευκος, write. From the same noun as a stem may be derived several verbs, having different terminations and different meanings; thus, δουλο-, δουλος, a slave ; δουλο-ω, I enslave ; δουλευω, I am a slave; πολεμο-, πολεμο-ς, war; πολεμε-ω, and πολεμιζω, Ι carry on war; woλεμo-w, to set in hostilities. Verbs may also be formed from verbs. There are three the idea conveyed by the primitive verb under certain modifications; these are called classes of verbs which set forth frequentative, inchoative, and desiderative. The frequentive are those verbs which denote a repetition of the act; the inchoative those which denote the incipiency or commence. ment of the act; and the desiderative are those which express a desire or inclination toward that which the primitive declares. These words are severally derived from frequentativus, a late Latin word, denoting repetition; inchoo, I begin; and desidero, I feel the want of, I wish for. 1. Frequentatives. Frequentatives are formed partly from the unchanged stem by means of the terminations αζω, ίζω, νίζω, partly by the conversion of the stem-vowel into o with the termination sw, or by the lengthening of o into w, the termination aw being added; e. g. |