It will, we think, be admitted, that in all countries and conditions in which men have been found, there exists a perception of a difference in the moral character of actions; that is, some things are accounted wrong, which ought not to be done, and some right which ought to be done. fundamental truths, there has never been any difference of opinion. It is not meant that all men distinctly think of these primary truths in morals; for many are so inattentive, or so much occupied with sensible objects, that they can scarcely be said ever to reflect on the subject of moral duty. Again, it has never been pretended as being a matter of But let an act of manifest injustice be performed before their fact, that between men of different countries there is a total eyes, and among a thousand spectators there will be but one difference in the opinions entertained respecting what is right opinion, and but one feeling. If a strong man, for example, and what is wrong. A few cases only of difference are alleged, violently takes away the property of one weaker than himself, in which this discrepancy is observed; but in regard to those and for no other reason than because he covets it, all men will actions which are reckoned good or evil, there is a general condemn the act. So, if any one who has received from agreement. As to those in which there seems to be a funda- another great benefits, not only refuses to make any grateful mental difference, an explanation will be given hereafter. No return, but, on the contrary, returns evil for good, all men will nation, or tribe, or class of mankind has ever held that it is a agree in judging his conduct to be wrong. All intuitively virtuous and proper thing to do injury to men, or that there is discern that for a ruler to punish the innocent and spare the no more harm in taking away life than in preserving it. It guilty, is morally wrong. It is not true, in fact, that there is has never been held that ingratitude-though everywhere no agreement among men as to the fundamental principles of common in practice-is a commendable thing; or that deceit morals. Their judgments on these points are as uniform as and fraud are as praiseworthy as honesty and fair dealing. on the axioms of mathematics; as in their agreement that the There is in every country a difference made in the estima-starry firmament is grand and beautiful; yea, as uniform as tion of the character of men, derived from the course of their concerning the greenness of the grass, or the varied colours of conduct. Some men are reckoned good in the public estima- the rainbow. tion, while others are considered wicked; the former obtain esteem, the latter are despised. That course of conduct which secures a good reputation, does not in any country consist of actions which we consider wicked, but of actions which in all countries are considered praiseworthy; and men have never obtained a bad character by a course of good behaviour. It is also important to observe, that the conduct of a people is not a fair test of the internal state of the mind, as it relates to morals. We know that individuals often pursue a course of conduct, which in their serious moments they condemn. Yet the power of temptation, and the habit of indulgence are such, that notwithstanding the convictions of conscience, they continue in a course of evil-doing. It would be a very inconclusive inference to determine from their habitual conduct, that they acted in accordance with the dictates of conscience. And what is true of individuals, may be true of nations and tribes. Those customs which they have received from their forefathers, may not meet with the approbation of their moral sense, and yet such is the force of an established custom, that they go on in the way in which they were brought up. Mr. Locke, in his zeal to disprove the existence of innate truths, attempts to render uncertain some of these first truths of morals. When we go beyond these first principles, we may expect to find men falling into grievous error respecting moral duty; and this often appears in their application of general principles to particular cases. Most men either do not reason at all, or reason badly, and draw from sound principles incorrect conclusions. For the most part, they receive implicitly what they have been taught; or they are governed in their opinions by the common sentiment; or they adopt as true what is most for their interest, or most agreeable to their feelings. And as men are often under the influence of feelings or passions which produce perturbation of mind, and so bias the judgment, it is easy to see how errors of judgment respecting moral conduct, in many cases, may spring up. And yet it is true, that there are primary truths in morals, in which all men agree, so soon as they are presented to the mind. As in other cases, by pursuing a course of sophistical reasonings, conclusions may be arrived at which are contradictory to these first principles, and this will produce perplexity; or even a kind of speculative assent may be yielded to such conclusions of ratiocination; but whenever it is necessary to form a practical assent in these cases is not a matter of choice, but of necessity. Bishop Berkeley thought he had demonstrated that there was no external world; and many others thought there was no flaw in his reasoning: but all these speculative sceptics were. nevertheless, practical believers in the real existence of external objects. Atheistical and infidel philosophers have often endeavoured to prove that there is no intrinsic difference between right and wrong, and some of them probably persuaded themselves that this opinion was true; but these very men, when an act of great injustice towards themselves or friends was committed, could not but feel that it was morally evil; and when they saw an act of disinterested benevolence performed, they could not but approve it as morally good. But a more satisfactory explanation of those facts, in which men seem conscientiously to go contrary to the fundamental principles of morals, is, that the principle on which they act is correct, but through ignorance or error they make an erro-judgment, the belief of intuitive truths must prevail. Our neous application of it. When parents murder their own female children-a thing very customary in China-it is on the principle that they will be subject to more misery than happiness in the world; and therefore it is doing them a favour. Here, the general principle is correct-that parents should consult the best interests of their offspring-but the mistake is in the application. The same may be said of the practice of exposing aged parents, when they become incapable of enjoying the world. As to those acts of cruelty which the Pagans perform in their religious services (the wife committing herself to the flames with the body of her deceased husband, children voluntarily thrown into the Ganges, or persons devoting their own lives by falling under the car of Juggernaut), they are performed on the principle that what God requires, or what pleases him, or what will secure happiness for ourselves or friends, should be done. It is true that the will of God should be obeyed, whatever sacrifice he may require; their error is in thinking that such sacrifices are required by Him. HOW FAR ALL MEN ARE AGREED IN THEIR MORAL JUDGMENTS. As the subject of morals is very extensive, and particular cases may be complicated, and as men are not only ignorant, but prejudiced by the errors received in their education, it is no more wonderful that they should adopt different opinions on these subjects than on other matters. That, however, which is true in regard to every department of human knowledge, is doubtless true in regard to the science of morals, There are certain self-evident truths, which are intuitively perceived by every one who has the exercise of reason, as soon as they are presented to the mind. In regard to these of a, i.e. that a is to be resolved into three equal factors; Obs. From the manner of performing evolution it is for a×a×a = 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. 1. Required the cube root of ao. Ans. a2. 2. Required the cube root of a or a1. Ans. a3, or 3√ a, For a3 X a3 × a° or 3va X sva X3√a=a. 3. Required the fifth root of ab. 4. Required the nth root of a2. 5. Required the seventh root of 2d-x. 6. Required the fifth root of (a-x)3. 7. Required the cube root of a1. 8. Required the fourth root of a1. 9. Required the cube root of a 10. Required the nth root of xm. 11. Required the third root of af. 12. Required the fourth root of x8. 13. Required the second root of an. 14. Required the fifth root of d3. 15. Required the eighth root of a3. 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. 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. The other 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. 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 the given quantity. 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 do, 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 into each other. 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 x is ±3. 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 square root of a is neither a nor-a. For+ax+a+a2. And ax-a=+a2 also. An even root of a negative quantity is, therefore, said to be impossible or imaginary. -- 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 ỏ, 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 2, and the root of y2, 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+b2. 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 x2 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 a2 + a ++ 4. Find the square root a2+fa +1. 5. Find the square root of a2 + ab + 2ab 6. Find the square root of a2 + + 2 A root whose value cannot be exactly expressed in numbers is called a SURD, or irrational quantity. Thus 2 is a surd, because the square root of 2 cannot be expressed in numbers with perfect exactness. In decimals, it is 1·41421356 nearly. Every quantity which is not a surd is said to be rational. By RADICAL QUANTITIES is meant, all quantities which are found under the radical sign, or which have a fractional index. REDUCTION OF RADICAL QUANTITIES. CASE I. To reduce a rational quantity to the form of a radical without altering its value. Raise the quantity to a power of the same name as the given root, and then apply the corresponding radical sign or index. 1. Reduce a to the form of the #th root. Over this, place the radical sign, and it becomes ny an. out any alteration of its value. For ny anana. 2. Reduce 4 to the form of the cube root. 3. Reduce 3a to the form of the 4th root. 4. Reduce ab 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 a b4 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. (1.) Reduce the indices to a common denominator. (2.) Raise each quantity to the power expressed by the numerator of its reduced index. 9. leduce at and (3.) Take the root denoted by the common denominator. 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 63. 3rd. The root denoted by the common denominator is the th. The answer, then, is (a) and (b2)14. The two quantities are thus reduced to a common index, without any alteration in their values. For aal, which = (@3)1a ̧ 1 m 1 23. Reduce am and bm to others with the common index 7. 24. Reduce a,, and 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 betacen 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. 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. 3. Reduce 52 and 63 to others with the common index . ti duóli, thou grievest si duble or dóle, he grieves ci dogliamo, we grieve vi doléte, you grieve si dólse, he grieved ci dolémmo, we grieved Mi dorrò, I shall or will grieve si dolgono or dogliono, they ti dorrúi, thou wilt grieve grieve Imperfect. Mi doléva or dolea, I grieved ti dolevi, thou grievedst si doléva or dolea, he grieved ci dolevámo, we grieved vi dolevate, you grieved si dolévano, they grieved Indeterminate Preterite. Mi délsi, I grieved ti dolesti, thou grievedst si dorrà, he will grieve 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.] Dubliti, grieve (thou) 4. Reduce a2 and aa to others with the common index.si dolga, let him grieve 5. Reduce 98 to its simplest form. 6. Reduce 243 to its simplest form. dogliamoci, let us grieve dolétevi, grieve (ye or you) si dolgano, let them grieve SUBJUNCTIVE MOOD. Che mi dolga or dóglia, that I may grieve che ti dolga or dóglia, that thou mayst grieve che si dolga or doglia, that he may grieve che ci dogliamo, that we may grieve che vi dogliate, that you may grieve che si dolgano or dógliano, that che si doléssero, that they might they may grieve So conjugate Condolérsi to complain. | Ridolére, to lament again. Observation.-Dolere signifies, also, to ache, and then it is used impersonally. Examples : Mi duble la tésta, my head ach s, or I have the headache. Mi dolgono i dénti, my teeth ache, or I have the toothache, V. Dovere, to owe. INFINITIVE MOOD. Simple Tenses. Compound Tenses, Present: dovere, to owe Past: avére dovuto, to have owed SUBJUNCTIVE MOOD. Present Gerund: dovendo, Past Gerund: avendo dovuto, Present. Imperfect. owing having owed Caglia, it may matter Calésse, it might matter. Past Participle: dovuto, owed giacé'te, you lie down Imperfect. Giace va or giace'a, I was lying down giace'vi, thou wast lying down giacciamo or giaciámo, we lie giace'va, he was lying down giacevamo, we were lying down ruto, seemed Present. Pájo, † I seem pári, thou seemest páre or pár, he seems pajámo, we seem parete, you seem pájono, they seem Imperfect. INDICATIVE MOOD. Pare'va or pare'a, I seemed seemed they Indeterminate Preterite. Párvi or pársi, I seemed giacciono or giáciono, they lie giacevano, they were lying parve or párse, he seemed down down pare'mmmo, we seemed pare'ste, you seemed párvero or pársero, they seemed Future. Parrò, I shall or will seem parrái, thou wilt seem parra, he will seem parre'mo, we will seem parre'te, you will seem parránno, they will seem Conditional Present. Parre'i or parría, I should or would seem parre'sti, thou wouldst seem parrebbe or parría, he would seem parre'mmo, we would seem parreste, 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 páro, pariámo, and parono, instead of pájo, Some Italians think that there is no imperative; but as excellent gram- pajámo, 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. |