188 



THE POPULAR EDUCATOR. 



participle, when in Spanish it is governed by another verb ; 

 as 



La oimos cantar, tee heard her | Le vi correr, I saw him run, or 

 singing (to sing). | running (to run). 



In Spanish the gerund is employed in the sense of the 

 present participle in English ; as 



Queriendo seguir dispensando- 

 )os, planto arboles frutales, desiring 

 to continue dispensing them, he planted 

 fruit-bearing trees. [speaking. 



Maria esta hablando, Mary is 



Estando enfermo el presidents, 

 the president being sicfc. 



Habiendo conspirado unos cabal- 

 leros, some cavaliers having con- 

 spired. 



In sentences such as Charity is increased by cultivating it, 

 it is allowed in Spanish to use either the gerund without the 

 preposition, or the infinitive mood preceded by the preposition 

 con; as 



Nuestros deberes se hacen agra- Our duties are rendered agreeable 

 dables cumpliendolos; or, Nuestros by performing them, 

 deberes se hacen agradables cou 

 cnmplirlos, 



Instead of the gerund of the past, the gerund of the present 

 is sometimes employed, preceded by the preposition en ; as 



En oyendo esto, salio para Bos- On hearing this, he set out for Bos- 

 ton ; or, Habiendo <5ido esto, salio ton / having heard this, he set out 

 para Boston, for Boston. 



The gerund in Spanish is often employed in a manner that 

 requires the adverb while to be used in translating it into 

 English ; as 



El quo vive en deleites, viviendo He tcho lines in pleasures, while 

 esta muerte, living is dead. 



The past participle is indeclinable when used to form the 

 compound tenses with the auxiliary verb haber ; as 



Las mugeres han hablado, the 1 Ella ha hablado, she has spoken, 

 women have spofcen. 



When the past participle is used with any other verb than 

 haber, it is declinable ; as 



Ella ha sido oeducida, she has 

 been deceived. 



Mis caballos estan lastimados, 

 m>/ 7iorses are injured. 



Mis bijas se ballan molestadas, 

 my daughters find themselves mo- 

 lested. 



Mis hermanas quedan satis- 

 fechas, my sisters remain satis/ted. 



Ellas van satisfechas, they go 

 satisfied. 



Ella anda pasmada de mis pala- 

 bras, she walks enraptured with my 

 words. 



Tiene escritas tres cartas, Tie has 

 three letters written. 



Lleva escritas tres cartas, he 

 carries written three letters. 



These last examples, it will be perceived, require the partici- 

 ple to agree with the noun governed (cartas). Tener and llevar 

 are, as above, sometimes used as a kind of auxiliary verbs, and 

 can always be rendered by have ; thus each of these examples 

 may be translated, he has written three letters. 



The past participle is in Spanish used with a noun or pronoun 

 in the case absolute; thus, liallado means found, and absolutely, 

 being found; enviado means sent, and absolutely, being sent; 

 recibido means received and being received. In general, the 

 participle is placed before the noun of the case absolute, with 

 which it agrees (though the rules of Spanish construction admit 

 of its being placed after the noun) ; as 



Tormada Yera-Cruz, el General Fera Cruz being tafcen, General 

 Scott salio para Jalapa, Scott set out for Jalapa. 



TENSES OF THE INDICATIVE MOOD. 



The present tense expresses an existing state or an action 

 occurring at the time in which we are speaking ; as 



Mi hermano escribe, my brother I Estas doncellas son amadas, 

 '.writes. | these maidens are beloved. 



The verb estar can be used with the gerund in Spanish, as in 

 English the verb to be with the present participle ; as 



Joan esta leyendo, John is read- 



ing. 



Ellos estan cantando, they are 

 singing. 



The verbs ir (to go) and venir (to come) do not admit of the 

 verb estar coming before their gerund as in the above rule. 

 Thus we cannot say in Spanish, yo estoy yendo and yo estoy 

 viniendo, but yo voy and yo vengo (I go and I come), I am going 

 and I am coming. 



The imperfect tense is used to express what is past, and, at 

 the same time, present, with regard to something else which is 

 past : that is. it is a past tense which was still present at the 

 time spoken of. It may always be employed in Spanish when 

 in English the word was can be used with the present participle, 



or used to can be employed with the verb, or when we speak of 

 habitual actions ; as 



Cervantes era un escritor ele- ' Ella ascribia entonces, she was 



gante, Cervantes was an elegant 

 writer. 



Neron era un tirauo, Nero was 

 a tyrant. 



writing then. 



Seneca razonaba bien, Seneca 

 reasoned well. 



Ellos marchaban por las calles 



Cuando fui niiio, hablaba como ' cuaudo los vimos, they were march- 



nino, when I was a child, I spoke as 

 a child. 



ing through the streets when we saw 

 them. 



It is evident that Seneca reasoned well means Seneca used to 

 reason (or ivas accustomed to reason) ^vell. 



The perfect definite tense shows the action or being affirmed 

 by the verb to be completed, at a time of which nothing more 

 remains, often specified by an adverb or some other circum- 

 stance expressed or understood ; as 



El presidente no le perdono, the 

 president pardoned him not. 



Diego vrvia cuando le vi, James 

 was living when I saw him. 



Escribio una carta ayer, he wrote 

 a letter yesterday. 



Recibio dos cartas la semana 

 he received (wo letters last 

 u-eck. 



Luego que Juan se lo dijo, llora- 

 ron, as soon as John told it to them, 

 they wept. 



PLANE TRIGONOMETRY. IV. 



XVIII. Application of the foregoing Formulas. It will be 

 readily seen how the power to work out numerical values for 

 functions of different angles is extended by the results of the 

 last few sections. We may now obtain values for the half or 

 third, or for twice or three times any of the angles whose values 

 were computed geometrically in Section V., and for any com- 

 binations arising by addition or subtraction of angles so 

 calculated. In this way, by steps which cannot be followed 

 here, the entire table of natural sines and cosines has been con- 

 structed, and by means altogether foreign to this treatise the 

 corresponding logarithms have been worked out for every degree 

 and minute from to 90 (see, amongst other works, Galbraith 

 and Haughton's Mathematical Tables). Moreover, the nume- 

 rous formulae derived from the "four fundamental formulae" 

 are of great use in helping us to simplify trigonometrical 

 expressions, and to change them into forms more suitable for 

 logarithmic calculation, or otherwise more convenient to deal 

 with. The following cases, given as examples, show how 

 apparently formidable expressions can be turned into simple 

 ones, easily solved, by mere knowledge how to make use of the 

 formulae : 



1. Reduce ^ -' to a single trigonometrical function, 

 sin. A 



By (63) and (60) 



1 - cos. A _ 2 sin. 2 j A __ sin. i A 

 sin. A 2 sin. A cos. A A cos. A 



i A 

 = tan. 4 A. 



2. Simplify 1 -+ c t \ A . By (25) and (52) 

 2 cot. A 



1 + cot. 2 A _ cosec. 2 A _ _ 



2 cot. A 2 cot. A sin. 2 A 2 cos. A 2 sin. A cos. A 



1 



sin. 2 A 



= cosec. 2 A. 



3. Reduce cosec. A + cot. A to a single function. By (62) 

 and (60) 



1 . cos. A 1 + cos. A 2 cos. 2 A 



Cosec. A + cot. A = - . + - - = '- s = 



sin. A sin. A sin. A 2 sin. f A cos. i^. 



sin. 2 A 

 4. Express tan. A + cot. A by a single function. 



, . sin. A ,cos. A sin. 2 A+ cos. 2 A. 1 



Tan. A + cot. A = r + i ~ ~7 i 



cos. A sin. A sin. A cos. A sm.Acos.A 



5. Reduce 



2 sin. A cos. A 

 sin. A 



= 2 cosec. 2 A. 



sin. 2 A 

 to a single function. This is similar to 



1 + cos. A 



Sin. A _ 2 sin. A cos. i A _ sin, j A _ 

 -j- cos. A 2 cos. 2 -| A cos. A 



