at an angle of 120°, because we always prefer to make use of Before proceeding to work this problem, we wish to give the the angle formed by the nearest approach of the projection to student some directions about the scale. In this case we the line of our position, or the picture plane. have given the representative fraction of the scale, and not 4th. Again, suppose an inclined shutter, or a roof which is the number of feet to the inch. It is a common practice united horizontally with a wall, is said to be at an angle of 40° with architects and engineers to name the proportion of the with the wall, the shutter or roof would be at an angle of 50° scale upon which the drawing is made, in the manner we with the ground. have done here, leaving the scale to be constructed if neces. All this will be very evident if we consider that “if any num- sary. The meaning of the fraction is is that unity is divided ber of straight lines meet in a point in another straight line on one into the number of equal parts expressed by the denomi. side of it, the sum of the angles which they make with this straight | nator. Thus a scale of feet is signifies that one standard line, and with each other, is equal to two right angles.” (See foot is divided into 48 equal parts, each part representing a Lessons in Geo- foot on paper, the metry, } Wol. I., result is ; inch page 156.) There- -- - to the foot. It fore (Fig. 67), if A Fig. 70. f also means that is 30° with the the original ob. PP, and B 90° with E. ject, whether a A, then B will be building or piece 60° with the PP, of machinery, is the whole making - ar 48 times larger two right les. than the drawing With re to ^ t- which, represent the last supposi- it. If the scale tion, we shall see had been written, that the lines of yards #, it would the wall, the roof DM m be the same as : or shutter, and inch to represent the ground, form o a yard. The way a right-angled tri- to arrive at this is angle, the three y as follows:– interior angles of - inches. . . which are together -h it. is of *=# inch to ground is 90°, and & The above method the shutter or roof - of stating the 40° with the wall, scale ought to be the shutter will be understood by at an angle of 50° every one engaged with the horizon r T- lo upon plan-draw: (Fig. 68). Conse- r ing. quently, this angle To return to the of 50° must be problem. The constructed for the principal cons. vanishing line, and deration relates to the subject treated the shutter. The as an inclined inclination maybe plane. (See Pro- upwards, at an an blems XXXI., gie of foowith the XXXII., and wall, or it may XXXIII.) From downwards at all this we deduct same angle. W. a rule for finding will representbo vanishing points cases. First, wh for lines or planes which are stated to be at given angles with other lines or planes not inclined do wards. Draw to HL, which is 4 fe from the ground Fig. 67. line; from Ps draw parallel with the a perpendicular * picture plane:— E.; this will be thi When the sum of radius for drawin the two angles of the given bbjects is greater than a right angle, the semicircle meeting the HL to determine DE' and no it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall ve", and its distano der is the extent of the angle sought. This will explain the re- point Dve"; also find the ve” by drawing a line from E to vo sults of the first, second, and fourth suppositions above. at a right angle with the one from E to wr', because if to When two angles of the given objects are together less than a shutter had projected from the wall in a horizontal position, t right angle, the sum will be the angle sought. This answers to would have vanished at ve”; that is, if it had been perpen the third supposition. We now propose a problem to illustrate dicular or at right angles with the wall. In short, the *. our remarks about the wall and the shutter. point for the horizontal position of a line must always be fo PROBLEM XLI. (Fig. 69).-A wall at an angle of 40° with whether the line retires to it horizontally or not, because the vi our position is pierced by a window of 4 feet 3 inches high and for an inclined retiring line is always over or under the vi 4 feet broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would has angle of 40° with the wall; the window is 5 feet from the retired if in a horizontal position. (See Prob. XXXI., Fig. 53. ground, and its nearest corner is 4 feet within the picture; other | Consequently, the vanishing point for an inclined retiring lin conditions at pleasure. Scale of feet #. is found by drawing a line from, in this case, the DVF, accord ing to the angle of inclination, to where it cuts a perpendicular line drawn through the ve”; thus we find its vanishing point, whether its inclination be downwards or upwards; therefore draw a line from DVP", at an angle of 50° with the HL, cutting the perpendicular from ve” at vp", the vanishing point. We hare made the nearest corner of the window 2 feet to the left of the eye, represented by the distance i to b; a line from h must be ruled to Ps, upon which we wish to cut of 4 feet to find a, the tearest point within; a line from c, which is 4 feet from ł, must be drawn to DE', and where it cuts the line *Ps in a is the point required. Draw the perpendicular a hom. Draw from DVF'through a top; make pr equal to the width of the window. Draw back again from r, cutting w?" ins; draw the perpendicular st; the base of the window is drawn from f, on the line of contact, 5 feet from the ground, to the ve"; the height of the window, | feet 3 inches, is marked from f to e : a line from e to ve", miting the perpendiculars from a and s in * and t, will give the top of the window. The opening of the window is on th on. Now we must draw wards, establishing its VP above the eye or HL.) Consequently, we must draw the vanishing line for the ve" downwards from DVP2. The sides of the shutter, t w and m v, must be drawn in the direction of ve", and cut off from DvP", first by drawing a line through t to y; make y a equal to the length of the shutter; draw from a to Dvip", producing w. All the early part of the problem, relating to the wall and windows, and the remaining lines w v and t m, will be but a repetition of the shutter under the first position. We can prove the truth of this method of drawing the perspective inclination of a plane by another method. Draw the right angle c a d (Fig. 68); make a b equal to the length of the shutter, and at an angle of 40° with a c or 50° with a d, draw b c parallel to a d: a c will be equal to the height of b above a. This must now be applied to Fig. 70. Draw a line from VP? through t to e on the line of contact; make ef equal to the height of b above a, viz., c a (Fig.68). Draw from fback to ve”; it will be found to cut the corner of the shutter in w, proving by both methods that t w is the perspective length of the further side of draw a PP across the paper in such a position with the plan, that by drawing visual rays,the picture-plane we have chosen may receive the view we wish to take of it. Suppose A (Fig. 71) is ** a line to the d * of contact, meet*it in y; make y z * to the length of the shutter, the same as the length of the *w; draw from a back, again to Dve", cutting t w in w; ** wo, directed by ve", and v m directed by ves. We will now draw the shutter at the same angle with the *!, but inclined upwards from it (Fig. 70). The important in working the problem under these conditions arises o the upward inclination of the shutter from the wall, but *d downwards to meet the wall. This last view of the Poition of the shutter is the proper one for our purpose, because *little consideration we shall perceive that it is a retiring : but downwards; therefore its vp is below the eye or H.L. * the former case the shutter was a retiring plane, but up the plan of a building, and we wished to have two views of it— one taken with an end and front in sight, the other with a view of the front and the opposite side—we should then place the PP at such an angle with the side or front as might be considered to be the best for our purpose. PP" would receive the visual rays from the front and the end B; PP” would receive those from the front and the end c. In short, any line may be drawn which represents the PP at any angle with the plan, or opposite any side we may wish to project. This will give a very useful illustration of the way to treat a subject when its proportions are given, as is frequently the case, without any reference to the view to be taken of it; in other words, the angle it forms with the picture-plane. at an angle of 120°, because we always prefer to make use of Before proceeding to work this problem, we wish to give the the angle formed by the nearest approach of the projection to student some directions about the scale. In this case we the line of our position, or the picture plane. have given the representative fraction of the scale, and not 4th. Again, suppose an inclined shutter, or a roof which is the number of feet to the inch. It is a common practice united horizontally with a wall, is said to be at an angle of 40° with architects and engineers to name the proportion of the with the wall, the shutter or roof would be at an angle of 50° scale upon which the drawing is made, in the manner we with the ground. have done here, leaving the scale to be constructed if neces. All this will be very evident if we consider that “if any num- sary. The meaning of the fraction is is that unity is divided ber of straight lines meet in a point in another straight line on one into the number of equal parts expressed by the denomi. side of it, the sum of the angles which they make with this straight nator. Thus a scale of feet is signifies that one standard line, and with each other, is equal to two right angles.” (See foot is divided into 48 equal parts, each part representing a Lessons in Geo- foot on paper, the metry, W., Wol. I., result is ; inch page 156.) There- - - - to the foot. It fore (Fig. 67), if A Fig. 70. f also means that is 30° with the 2^ the original ob. PP, and B 90° with E. ject, whether a A, then B will be 60° with the PP, the whole making building or piece of machinery, is 48 times larget two right les. than the drawing With re to which represents the last supposi- it. If the scale tion, we shall see had been written, that the lines of yards A, it would the wall, the roof be the same as : or shutter, and inch to represent the ground, form a yard. The way a right-angled tri- to arrive at this is angle, the three as follows:– interior angles of which are together angles. Therefore, se” * , ; as the angle of the is of * = #.o wall with the J.-ground is 90°, and The above method the shutter or roof 40° with the wall, the shutter will be at an angle of 50° with the horizon (Fig. 68). Conse of stating the scale ought to be understood by every one engaged upon plan-draw: ing. quently, this angle To return to the of 50° must be problem. The constructed for the principal consi vanishing line, and deration relatest: the subject treated the shutter. The as an inclined inclination mayb plane. (See Pro- upwards, at an an blems XXXI., gle of 40° with thi XXXII., and XXXIII.) From all this we deduct a rule for finding vanishing points for lines or planes which are stated to be at given HL, which is 4fe angles with other from the ground lines or planes not Fig. 67. line; from Ps dra parallel with the a perpendicular picture plane — E; this will be to When the sum of radius for drawin the two angles of the given bbjects is greater than a right angle, the semicircle meeting the HL to determine D.E." and Dr. it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall ve", and its distan der is the ertent of the angle sought. This will explain the re- point DvP"; also find the ve” by drawing a line from E to v. sults of the first, second, and fourth suppositions above. at a right angle with the one from E to ve", because if to When two angles of the given objects are together less than a shutter had projected from the wall in a horizontal position, right angle, the sum will be the angle sought. This answers to would have vauished at vp”; that is, if it had been perpe the third supposition. We now propose a problem to illustrate dicular or at right angles with the wall. In short, the vanishiour remarks about the wall and the shutter. point for the horizontal position of a line must always be fou PROBLEM XLI. (Fig. 69).-A wall at an angle of 40° with iwhether the line retires to it horizontally or not, because the our position is pierced by a window of 4 feet 3 inches high and for an inclined retiring line is always over or under the 4 feet broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would ha angle of 40° with the wall; the window is 5 feet from the retired if in a horizontal position. (See Prob. XXXI., Fig.5: ground, and its nearest corner is 4 feet within the picture; other Consequently, the vanishing point for an inclined retiring li conditions at pleasure, Scale of feet to is found by drawing a line from, in this case, the Dvro, acco. wall, or it may b downwards at th same angle. W will representbot cases. First, whe inclined door wards. Draw th ing to the angle of inclination, to where it cuts a perpendicular line drawn through the vr"; thus we find its vanishing point, whether its inclination be downwards or upwards; therefore draw a line from DvP*, at an angle of 50° with the HL, cutting the perpendicular from vp” at vp”, the vanishing point. We have made the nearest corner of the window 2 feet to the left of the eye, represented by the distance i to b; a line from b must be ruled to Ps, upon which we wish to cut of 4 feet to find a, the nearest point within; a line from c, which is 4 feet from b, must be drawn to DE', and where it cuts the line b Ps in a is the point required. Draw the perpendicular a hom. Draw from DVP"through a top; make pr equal to the width of the window. Draw back again from r, cutting bvr" in s, draw the perpendicular st; the base of the window is drawn from f, on the line of contact, 5 feet from the ground, to the we'; the height of the window, 4 feet 3 inches, is marked from f to e : a line from e to ve", cutting the perpendieulars from a and s in r; and t, will give the top of the window. The opening of the window is on th n. Now we must draw wards, establishing its vp above the eye or HL.) Consequently, we must draw the vanishing line for the ve" downwards from DvP2. The sides of the shutter, t w and m v, must be drawn in the direction of ve", and cut off from DvP*, first by drawing a line through t to y; make y a equal to the length of the shutter; draw from a to Dve", producing w. All the early part of the problem, relating to the wall and windows, and the remaining lines w v and t m, will be but a repetition of the shutter under the first position. We can prove the truth of this method of drawing the perspective inclination of a plane by another method. Draw the right angle c a d (Fig. 68); make a b equal to the length of the shutter, and at an angle of 40° with a c or 50° with a d, draw b c parallel to a dj a c will be equal to the height of b above a. This must now be applied to Fig. 70. Draw a line from ve” through t to e on the line of contact; make ef equal to the height of b above a, viz., c a (Fig.68). Draw from f back to ve”; it will be found to cut the corner of the shutter in w, proving by both methods that t w is the perspective length of the further side of draw a line to the Hae of contact, meeting it in y; make y z equal to the length of the shutter, the same as the length of the window; draw from a back, again to Dve", cutting t w in w; draw to v, directed by ve", and v m directed by ves. We will now draw the shutter at the same angle with the wall, but inclined upwards from it (Fig. 70). The important iifference in working the problem under these conditions arises from the upward inclination of the shutter from the wall, but factoried downwards to meet the wall. This last view of the position of the shutter is the proper one for our purpose, because after a little consideration we shall perceive that it is a retiring orae, but downwards; therefore its ve is below the eye or HL. Ia the former case the shutter was a retiring plane, but up -מט 0 STEYS, PERSOXAL ENDINGS. LESSONS IN GREEK.-XXV. The simple syllabic augment is found in only the indicative mood; the reduplicative extends through all the moods. The CONJUGATION.-PRELIMINARY NOTIONS. simple syllabic augment is used with the imperfeot tense and LET us take the word cavoaunv to illustrate what was said in with the aorist. The reduplicative angment is used with the the last lesson. The word signifies I loosed myself, I untied or perfect tense, the pluperfect tense, and the third future, someunbound myself. Now suppose that I unbound myself was times called the paulo-post-future. If, however, the verb begins written as though it formed one word, as thus :— Iunboundmy, with a vowel, the perfect and the pluperfect have, instead of self. If we mark off the several elements of this compound by the reduplicative, merely the temporal augment. The pluperhyphens, and assign names to the several parts fect has a double augment, inasmuch as it prefixes the simple Personal Prefix. Adverbial Prefix. Verbal Stem. Personal Suffix. augment e to the reduplicative te, etc.; for instance, ETETVDELY. I bound Fuller details will be given heroafter. we may have some idea how the Greek form above presented CHARACTERISTIC LETTERS. has been produced. Here it is divided, and the parts named :Augment. Root. Aorist Stem. Middle Personal Ending. We have used previously the terms pure verbs. This is one λυ unu. class into which verbs are divided. Verbs are divided geneIt is thus seen that the root of the form is nv. This is called rally into classes, according to the characteristic letters of the the root, because it remains permanent under all the modifica- present tense, or the stem of the present tense. The letter tions. Thus it is found in Avow, in Avgouevos, eAvony, etc. By called the characteristic letter: thus, in Auw, the v is the charac which stands immediately before the w of the present tense is prefixing certain letters to Av, and by adding certain letters to Av, we get all the varieties of form and signification. Thus, if teristic of the verb; in TviTW, the 7 is the characteristic of we want to say I loose, we add w, as Au-w; if we want to say verb. If the characteristic is a vowel, the verb is called pure, the verb; and in otellw, the 1 is the characteristic of the they loosed, we prefix € and add cav, thus, e-Au-ray. The prefixes and suffixes by whose aid the root is thus modified may be c.9., \w; if the characteristic is a consonant, the verb is termed formative syllables. A knowledge of these formative called mute, e.g., TUNTW; if the characteristic is a liquid, the verb syllables, combined with a knowledge of the several roots, is is called liquid, e.g., oteldw, I send. Thus there are three kinds of verbs. necessary for a correct knowledge of the grammar of the verbs. Pure. Liquid. Tilaw, I honour. Tpubw, I rub. and the stem. The root of a verb is the verb reduced to its pauw, I show. ultimate or most simple form. It agrees with the stem in being FLEXIONAL TERMINATIONS. generally the stem of the present tense, active voice. But it differs from the stem, inasmuch as it is one primitive form, inflexions, which mark the time (tense), the manner (mood), Another kind of characteristic letters or syllables are the and there are several stems—the stem of the present, the stem and the persons of the verb. Look at Avooual, I will loose of the imperfect, the stem of the perfect, etc. tense is that form which remains when the personal endings myself . Analyse it , and the parts will be found to stand thus: Root. Tense. Sign. Mood Sign. Person Sign. and the mood characteristics are taken away. The following Au feat. are the stems of the root and of several tenses of TUTTW, I strike. Here av is the root, o is the characteristic of the future, o of the indicative mood, and pai of the first person singular. Let Third Person. Second Person. us vary these forms a little. Root, Root, Tense Sign. Mood Sign. Person Sign. -El, he strikes; μεθα. . Imperfect Stem, ETUTT. -E, he was striking; -es, Here the sign of the indicative mood, o, has become on, to indi. First Aorist Stem, etry. -€, he struck ; cate the optative, and ual of the first person singular is changed Perfect Stem, TETVO. •Ehe has struck; -as, into ueda of the first person plural. Again, take e vrarto. Pluperfect Stem,' eTeTVD. -El, he had struck; els, Augment. Root. Tonse Sign. Mood Sign. Person Sign. That is to say, if to the present stem el be added, we get TUTTEL, which means he strikes; if to the pluperfect stem els be added, Be βουλευ Lial. we get et ETUDES, which means thou hadst struck. So, if from TUT- λυ. OL -as, du VTO. 0 βουλευ . uny. τετυφας we take away ας, we get the perfect stem τετυφ. If Augment. Root. Voico Sign. Person Sigr. we want to make the perfect stem into the pluperfect stem, we βουλευ mv. prefix the augment e, and make ETETUD. If, again, we wish to The tense sign, in union with the person sign, is termed the resolve Tetup into the root, we have to cut off the augment te, tense-ending. Thus in Avow the o is the tense sign, being the and change the aspirate o into the corresponding soft #, and so sign of the future, and ow is the ending of the future tense, obtain Tum. This, the root, can be raised into the present stem active voice, commonly called the first future active. The stem by affixing T-thus, TUAT. And Tynt may be changed into the of the verb, in connection with the tense sign and with the imperfect stem by prefixing the augment of that tense, namely, €. augment, is called the tense-stem. Thus, in εβουλευσα the tense-stem is eßovlevo—that is, the stem of the first aorist First of all, we must consider the augment or temporal prefix. active. We call the augment temporal, because its function is to denote GENERAL TABLE OF THE TENSE-ENDINGS. past time; and we call it a prefix, because it is put at the be Active. Middle. Passics. ginning of the root or stem. The augment is of two kinds : Present, •ομαι. . first, syllabic ; second, temporal. It is syllabic when it adds a Imperfect, -OV, syllable to the verb; it is temporal when it lengthens the initial Perfect, vowel of the verb. The syllabic augment is of two kinds, it is Pluperfect, simple or reduplicative. For instance, it is simple when it Aorist First, «σάμην, , merely prefixes & vowel, as in edelmov, I was leaving; it is re Future First, •σομαι, -θησομαι. duplicative when it doubles the initial consonant, as Aeluka : Aorist Second, -ouny, here e is called the simple syllabic augment, and ae the redupli Future Second, - nooual. cative. The syllabic angment is employed when the verb begins This arrangement places under the middle voice some tenses, with a consonant. If the verb begins with a vowel, the tempo- those marked with an asterisk, which are commonly ascribed ral augment is used, the vowels a and e being changed into n or to the passive voice. If the student bears in mind what was el, and i and ů (iota short and upsilon short) being changed said in the last lesson of the intimate relation of the two, he into i and ü: o is changed into w. In the same way, in verbs will see a ground for this diversity of view. beginning with the diphthongs al, el, ou are changed into ?, , the first vowel being changed into its corresponding long one, PERSONAL ENDINGS AND TOWEL SIGNS. and the : written underneath; av becomes nu. If a verb begins The personal endings are the terminations by which the raciawith p, the p is generally doubled, as Dettw, I throw, eppuntov. tions of person are indicated. They are closely connected wità THE AUGMENT. -ojny. * -pal. linu.* oa, -0, -OV, |