SONS IN ALGEBRA. 



Alo, the adjective* dignus and indignut take the ablative ; 







Virtus imituf.oiie noil ins ! i ilitfua et, 

 . u tourthi/ o/ imitation, nut u/ . 



The ablative absolute has already been explained Bat 



observe, the ablative absolute is an abbreviated sentence lieing 



an abbreviated sentence, it has a subject of its own. C'onse- 



. , its subject is different from the subject of the complete 



sentence of which it forms a part. There must, then, in thin 



tiou be two subjects, that of the ablative absolute and 



: tue full sentence. To exemplify these remarks, you 



may say, with both sentences complete 



1 Sentence. Second Sentence. 



Full Construction. ROIUBB reges exacti sunt, et cousules creati aunt. 

 One Sentence. 



Ablative Absolute. Uouue, regibus exactia, consults creati sunt. 



Here you see that the ablative absolute holds the place of 

 Hie first sentence, and is that sentence in an abbreviated form. 

 Yon alsc see that the ablative absolute construction involves 

 two subjects ; here, for instance, regibns and consules. Conse- 

 quently, there cannot be an ablative absolute construction 

 without two subjects. 



VERBS GOVERNED BY VERBS. 



Verbs may govern verbs. A verb governs another verb when 

 the second verb is dependent on the first. The simplest case of 

 verb-government is that which is expressed in the general form 

 or rule : 



One verb governs another in the infinitive mood ; as 



Ctesar maturof ab urbe proficisci, Casar haste-i* to go from the city; 

 where proficisci in the infinitive mood is governed by matnrat. 



The infinitive may be the subject of a verb as well as the 

 object; as 



Humanum eat errare, To err is human. 



Errare is the subject to cst ; in sense err are is nearly the same 

 as the noun error. 



A noun may accompany the infinitive, whether it is the sub- 

 ject or the object ; as 



SUBJECT. 



Homines errare nou mirum cst, 

 for men to err is not wonderful. 



OBJECT. 



Scio ducem ease fortem, 



I know that the general is brave. 



Should the infinitive be accompanied by an adjective, noun, or 

 pronoun intended to complete the meaning, the accompanying 

 word must agree with the subject, if it is to be taken with the 

 subject ; and with the object, if it is to be taken with the 

 object; as 



SUBJECT. OBJECT. 



Coapi tibi melestus ease, I Jubeo fe esse/orfem, 



I began to be troublesome to thee. \ I command thce to be brave. 



If the verb governs a dative case, the accompanying word or 

 words may be in the dative ; as 



Mihi negligent* ease non licet, J must not be negligent. 



We may exhibit a similar construction with an accusative : 



Me negligent*, ease non licet, It is not proper for me to be negligent. 



A few verbs govern a subjunctive mood. 



This construction must be distinguished from the construc- 

 tion in which a subjunctive mood is connected with a verb in 

 the indicative mood together with a conjunction. In the con- 

 struction now before us there is no conjunction ; at least none 

 is expressed, though in all probability the construetion is 

 elliptical ; as 



Magnum fac animum habeas, Take care to keep a high mind. 



That ut was originally used in such constructions may appear 

 from its still continuing to be occasionally employed ; as 



Cura wt valeas, Take core to keep v, !!. 



The verbs with which the construction is found are volo, 

 malo, nolo, euro, censeo, permitto ; also oro, quseso, rogo, 

 precor, postulor, peto, hortor, suadeo, moneo, mando, decerno ; 

 and again the imperatives fac, cura, having the force of our do, 

 or be sure to do, take care that you, etc. Volo, etc., may stand 

 in the subjunctive, having the force of a kind of softened com- 

 mand or wish ; as 



Diligens su velim. J should like you to be tndutstrtoui. 



Indeed both nonitructions occur ; aa may be eon, for example, 

 in the following sentences : 



Aecutattve. Lieut ThemutoeUm MM ofioium. 



It u permilted that Tktmutoeltt i* idU. 

 Dative. Themi'itocfi lir-uit MM otv,u>, 



It wot permitted to T/itmutocU* to It idle. 



Instead of the infinitive with the accusative, come verbs tnkft 

 also the indicative with quod ; as 

 Gaudeo to vaJere, or 

 Guuilco quod valet, / am glad that you art vtll. 



In this case the quod is the relative pronoun, the object to 

 the verb gaudeo, thus, I rejoice at this (namely) that yon are 

 well. Quod in such instances as the present is commonly called 

 a conjunction, and as a conjunction may it here be regarded ; 

 not the less is it traceable to the relative force which is its 

 essential meaning. Of our that and of our because [be (by) and 

 cause] similar remarks may be made. 



LESSONS IN ALGEBRA. XXXVIII. 



GEOMETRICAL PROPORTION AND PROUiiEiSION. 

 IF four quantities are in geometrical proportion, the product oj 

 the extremes is equal to the product of the means. Thus, 

 12 : 8 : : 15 : 10 ; therefore 12 X 10 = 8 X 15. Hence, 



Any factor may be transferred from one of the means to the 

 other, or from one extreme to the other, without affecting the 

 proportion. 



Thus, if a . mb -. : x -. y, then a : b : : mx : y ; for the product of 

 the means in both cases is the same. 



So, if na . b : . x . y, then a : b : : x -. ny. 



On the other hand, if the product of two quantities is equa 

 to the product of two others, the four quantities will form a 

 proportion, if they are so arranged that those on one side of the 

 equation shall constitute the means, and those on the other side 

 the extremes. Thus, since 6x12 = 8x9, then 6 : 8 : : 9 : 12. 



Corollary. The same must be true of any factors which form 

 the two sides of an equation. Thus, if 



(a + 6) X c = (d - m) X y, then a + b -. d - m -. -. y : e. 



If three quantities are proportional, the product of the 

 extremes is equal to the square of the mean. For this mean 

 proportional is, at the same time, the consequent of the first 

 couplet, and the antecedent of the last. It is, therefore, to be 

 multiplied into itself; that is, it is to be squared. 



Thus, 4 : 6 : : 6 : 9 ; therefore, 4x9 = 6x6. 



If a : b : : b : c, then multiply extremes and means, ac = 6*. 



Hence, a mean proportional between two quantities may be 

 found by extracting tlie square root of their product. 



If a : x : : x -. c, then j? = ac, and x = Vac. 



In a proportion, either extreme is equal to the product of the 

 means, divided by the other extreme ; and either of the means 

 is equal to the product of the extremes, divided by the other 

 mean. 



1. If a : b : : c : d, then ad = be. 



2. Dividing by d, a = be d. 



3. Dividing the first by c, b = ad c. 



4. Dividing it by b, c = ad 6. 



5. Dividing it by a, d = be o. 



That is, the fourth term is equal to the product of the second 

 and third divided by the first. 



N.6. On this principle is founded the rale of simple propor- 

 tion in arithmetic, commonly called the " Rule of Three." Three 

 numbers are given to find a fourth, which is obtained by multi- 

 plying together the second and third, and dividing by the first. 



The propositions respecting the products of the means and of 

 the extremes, furnish a very simple and convenient criterion for 

 determining whether any four quantities are proportional. We 

 have only to multiply the means together, and also the extremes. 

 If the products are equal, the quantities are proportional. If 

 the products are not equal, the quantities are not proportional. 



It is evident that the terms of a proportion may undergo any 

 change which will not destroy the equality of the ratios, or 

 which will leave the product of the means equal to the product 

 of the extremes. These changes are numerous, but they may bo 

 reduced to a few general principles. 



