310 



THE POPULAB EDUCATOR. 



intersection of the perpendiculars B K, G K, drawn, as before, at 

 right angles to the extremities, B and G, of the given straight 

 lines A B, G H. 



Next, let the given straight lines A B, c D be parallel to one 

 another. Through B and c (Fig. 99) draw F B x, c E, perpendicular 

 to A B, c D respectively. Join B c, and having taken a point, K, in 

 B F, so that B K shall be less than B c, draw K L through K, 

 parallel to B c, and cutting c E in L ; from L as centre, with the 

 distance L c, which is equal to B K, describe the arc c M, meeting 

 B c in M. Join i. M, and produce it in the same straight line 

 towards M, to meet FX in N. From N as centre, with the 



distance N B or N 

 M, describe the arc 

 B M. The given 

 straight lines AB, 

 CD are connected 

 by the curve BMC. 

 If, however, the 

 given straight lines 

 are not parallel, 

 but would meet if 

 one or both were 

 produced, as a H 

 produced meets B A 

 in A, forming the 

 small angle HAB, 

 draw, as before, F x 

 and o o at right 

 angles to A B and 

 o H respectively. 

 Take any point, K, 

 in B F ; make o P equal to B K, and join K p. Bisect K p in 

 Q. and draw Q R perpendicular to K p, meeting F x in B. Join 

 B p, and from p as centre, at the distance p o, describe the arc 

 G s, meeting B p in s. Then from the centre B, at the distance 

 BB or K8, describe the arc B s, completing the curve B s G, by 

 which the given straight lines A B, G H are connected. 



This problem exhibits a mode of construction useful to 

 engineers in laying out the curves of a railway ; to landscape 

 gardeners, in laying out walks and roads ; and to carpenters, 

 in forming curves to connect the straight edges of a piece of wood 

 by a curve, when they are either parallel to one another, or in- 

 clined to each other at a greater or less angle. 



At this point we bring to a conclusion our Lessons in Geo- 

 metry, in which we have explained, as clearly and as fully as 

 possible, the leading principles of the science on which all the 

 constructive arts are based. Of the practical value of geometry 

 to the artisan and mechanic we have already given many proofs. 

 It will not be too much to say that any one who has studied 

 these lessons carefully, and understands them thoroughly, has 

 not only rendered himself a scientific workman, but has ad- 

 vanced far on his way to become an architect or civil engineer, 

 or to enter any profession in which a knowledge of geometry is 

 an essential requisite. From these lessons the student will 

 find it of the greatest advantage to turn to " Euclid's Elements 

 of Geometry," in which he will find a conclusive proof of almost 

 overy construction that has been brought under his notice in 

 the preceding problems. 



Pig. 99. 



Inf. Pres. 



Latin. Amare, 



English.. to love. 



Latin. 

 English. 



Latin. 

 English. 



1 Sup. 

 Amatum, 

 to love. 



Gfcr. 



Arnandi, 

 of loving. 



Latin. 

 English. 



2 Sup. 

 Amatu, 

 to be loved. 



Gr. Ger. 



Amando, Amandum, 

 to loving, loving. 



Inf. Fut. 

 Amaturum esse, 

 to be about to love. 



Prcs. Part. 

 Amans, 

 loving. 



Ger. 



Amando, 

 by loving. 



Imp. 

 Ama, 

 love thou. 



Fid. Part. Act. 

 Amaturus, 

 about to lov. 



Inf. Perf. 

 Amavisse, 

 to have laved. 



LESSONS IN LATIN. XXIII. 



REGULAR VERBS. THE FIRST CONJUGATION. 

 ACTIVE VOICE. 



Example. Amo, I love. Chief Parts: amo, amavi, amatum, 

 amare. Characteristic letter, a. 



PABTS WITH THE COEEBSPONDING ENGLISH. 



Sub. Pres. 

 Amem, 

 I may love. 



Latin. 

 English. 



Ind. Pres. 

 Amo, 



I love, } 



I do love, or > 

 I am lowing.) 



I Put. 



Latin. Amabo, 

 English. I shall or actll lot-*. 



Ind. Imp. 

 Amabam, 

 I did love 



'"I 



ing.) 



Sub. Imp. 

 Amarem, 

 I might love. 



Iicas lovi 



2 Put. 

 Am:ivero, 

 I shall have loved. 



Latin. 



English. 



Jnd. Perf. 

 Amavi, 



Sub. Perf. 

 Amaverim, 



Ind. Pluperf. 

 Amaveram, 



Sub. Pluperf. 

 Amavissem, 



I have loved. I nay have loved. I had loved. I might have loved.. 



Before I proceed, I will explain these contractions : 

 Contractions. Emanation. 



Ind. Pres. Indicative Mood, Present Tense. 



Sub. Pres. Subjunctive Mood, Present Tense. 



Ind. Imp. Indicative Mood, Imperfect Tense. 



Sub. Imp. Subjunctive Mood, Imperfect Tense. 



1 Put, First Future Tense. 



2 Put. Second Future Tense. 



Ind. Perf. Indicative Mood, Perfect Tense. 



Sub. Perf. Subjunctive Mood, Perfect Tense. 



Ind. Pluperf. Indicative Mood, Pluperfect Tense. 



Sub. Pluperf. Subjunctive Mood, Pluperfect Tense. 



Inf. Prea. Infinitive Mood, Present Tense. 



Inf. Perf. Infinitive Mood, Perfect Tense. 



Inf. Put. Infinitive Mood, Future Tense. 



Imp. Imperative Mood. 



1 Sup. First Supine. 



2 Sup. Second Supine. 

 Pres. Part. Present Participle. 



Put. Part. Act. Future Participle, Active Voice. 



Ger. The Gerund. 



Having in the above corresponding parts given the Latin as 

 well as the English of several members of the verb, I need not 

 repeat them. I supply in full what remains. As I write for 

 young men and women rather than for children, I omit adding 

 the English in all the details of the persons ; for when you know 

 what is the first person, you will readily supply the rest : thus, 

 if the English of amaveram, the first person, is I had loved, 

 you know that the English of amaveras, the second person, is 

 thou hadst loved; and of amaverat, the third per&on, he had 

 loved ; so also in the plural. 



Instead of I might have loved, the sub. pluperf. may some- 

 times be rendered (put into English) by I would, I should, or I 

 could have loved. 



In the corresponding English words, I have given the nearest 

 approach to the several Latin parts. The student will do well 

 to adhere strictly to these meanings at first, though, as the 

 correspondence between the several Latin and the several 

 English parts is not entirely complete and constant, he will find 

 occasions when his English will appear scarcely idiomatic, or 

 strictly proper. He cannot, however, learn too soon, that in 

 few particulars are any two languages exactly correspondent. 

 Accordingly, for amo, I have set down what may be termed 

 three meanings namely, I love, I do love, and I am loving. 

 Here it is obvious that the English is more rich than the Latin, 

 inasmuch as it has three forms of the present tense indicative 

 mood, while the Latin has but one form. Having but one form, 

 the Latin cannot by a form indicate the variations of the English 

 present tense. Consequently, here is a want of strict corre- 

 spondence ; and here also is a source of doubt ; for we may ask, 

 what is the English equivalent of amo P is it, I love, or I do 

 love, or I am loving? 



After these remarks the student will know that it is with 

 some latitude that he is to take these 



COBBESPONDING LATIN AND ENGLISH SIGNS. 

 Ind. Pres. Sub. Pres. Ind. Imp. Sub. Imp. 1 Put. 2 Fui. 

 -bam, -rem, -bo, -ero, 



did. might. will, icill have. 



Ind. Pluperf. Sub. Pluperf. Inf. Pres. 

 -eram, -issem, -are, 



had. might have. to. 



English, to have. 

 2 Sup. Part. Pres. Put. Part Act. 



Latin. 

 English. do. 



Ind. Perf. 

 Latin. -i, 



English, have. 



-em, 



may. 



Sub. Perf. 



-eriin, 

 may have. 

 Inf. Perf. Latin, -isse; 

 Inf. Put. Imp. 1 Sup. 



Latin, -rum esse, 

 English, about to. 



ama, -um, 

 do. in order to. 



to. 



-ans, 

 -ing. 



rus, 



on the point of. 



Give yourself a thorough practice in these signs. Again and 

 again ask until you are perfect, what is the English sign of the 



