Page images

Cear. IV.



F. 276.

Bracciano ; the façade and cortile of the palace of Cardinal Titelleschi, now the hôtel palazzecio, at Cometo; the west front of the church of Sta. Maria in Strada, a most elaborate work in brick and terra-cotta, and the church of the dominicans, at Monza; all belong to the last period of Italian Gothic. The nave of the church of Sta. Maria delle Grazie, at Milan, is pointed and dated 1465, while the transepts and ehoir are thirty years later, and are renaissance work. The church of Sta. Maria Maggiore, at Città del Castello, be. longs to the 15th, but was finished in the 16th, century. The cbureh of San Agostino, at Ancona, is transitional ; like that at Montenegro, 1450; and that of the Madonna di Monte Luce, at Perugia. The last idea of Gothie art absorbed by the new

AL style, is seen in the Colleone etapel, 1475; and in the church of Sta Maria Maggiore, at Bergamo, where the sacristy, 1430, offers one of the earliest dated examples of the modern style. There is scarcely a street in Città della Pieve without bumerous cases of pointed door. ways and windows walled up

ELEVATION OF CATHEDRAL, COMO. to suit the return to what are commonly, but incorrectly, called classical notions.

625. Such are the chief structures in the northern half of Italy, of which a critic so highly esteemed as Prufessor Willis does not hesitate to affirm that, “ there is in fact no genuine Gothic building." The same author observes that, “it is curious enough that in the Neapolitan territory, in Naples especially, many specimens or rather fragments, of good Gothie buildings are to be found which were executed under the Angevine dynasty, 126F1435; with this exception I do not believe that a single unmixed Gothic church is to be found in Italy." Others follow his judgment, and accept, as specimens of imitative Gothic art, edifices which they themselves describe as impure and heterogeneous, and impressed with thu stamp of classical, romanesque, byzantine, and saracenic influences. To this praise of the churches may be added that of two or three palazzi at Naples; the campanili at Amalfi, and Velletri; the castles at Andria, Castellamare, and Teano, some houses of the 14th and 15th centuries, at Aquila, Popoli, and Solmone, with the aqueduct at the latter place; and the monastery of Sta. Catherina, at Galatino.

626. The cathedral at Trani must be regarded as falling within the ban under which the structures termed - Gothic,” in Sicily, are regarded by the purist in archælogy. The pointed byzantine style, which is called Siculo-Norman, lasted until 1282 ; it was transitional in the sense of receiving greater enrichment of a Greek character, until the end of the 14th century; and although further change began in the 15th century, taste did not take any decided direction until the establishment of renaissance art. M: Gally Kniglit, sbo investigated the indications presented in the great work published by Messrs. Hittorff and Zanth, says that " various novelties were attempted; sometimes the forms were circular, sometimes square, and soinetiines elliptic. Amongst other novelties, the pointed style of the north was introduced, with its projecting mouldings and a little of its tracery; but later in Sicily than anywhere else ; and though something of its true spirit is caught in the reconstructions of Maniaces, in Syracuse, yet in Sicily, it always appears an exotic." These facts seem, to Mr. Freeman, to prove incontestibly that the pointed style of Sicily, of that portion of western Christendom in which the systematic use of the pointed arch first occurred. is not Gothic even in the sense of being the most distant transition. A few churches and palaces at Palermo, Syracuse, and Taormina, of the 14th century; and in the same cities, with Girgenti and Messina, of the 15th century, would be nearly all that could be named as important examples before the renaissance was employed. The date, 1592, however, appears to be that of the elliptic arches, groined roof, and Aamboyant parapet at the entrance to the church of Sta. Maria della Catena, at Palermo.

627-873. We here, with regret, leave the subject, because we have already trespassed beyond the limits prescribed.







Sect. I.

GEOMETRY. 874. Geometry is that science which treats of the relations and properties of the boi daries of either body or space. We do not consider it would be useful here to notice history of the science; neither is it necessary to enter into the reasons which have indu« us to adopt the system of Rossignol, from whom we extract this section, otherwise than state that we hope to conduct the student by a simpler and more intelligible method those results with which he must be acquainted.

The limits of body or space are surfaces, and the boundaries of surfaces are lines, and t terminations of lines are points. Bounded spaces are usually called solids, whether occupi by body or not; the subject, therefore, is naturally divided into three parts,-lines, surfac and solids ; and these have two varieties, dependent on their being straight or curved.

875. Geometrical inquiry is conducted in the form of propositions, problems, and dema strations, being always the result of comparing equal parts or measures. Now, the pa compared may be either lines or angles, or both; hence, the nature of each method shou be separately considered, and then the united power of both employed to facilitate 1 demonstration of propositions. But the reader must first understand these Definitions. 1. A solid is that which has length, breadth, and thickness. A slab of marble, 1

instance, is a solid, since it is long, broad, and thick. 2. A surface is that which has length and breadth, without thickness. A leaf of pap

though not in strictness, inasmuch as it has thickness, may convey the idea of a surfa 3. A line is that which has length, but neither brcadth nor thickness.

As in the case a surface, it is difficult to convey the strict notion of a line, yet an infinitely thin lii

as a hair, may convey the idea of a line : a thread drawn tight, a straight line. 4. A point is that which has neither length, breadth, nor thickness. 5. If a line be carried about a point A, so that its other extremity

passes from B to C, from C to D, &c. (fig. 223.), the point B, in its revolution, will describe a curve BCDFGLB. This curve line is called the circumference of a circle. The circle is the space enclosed by this circumference. The point A, which, in the formation of the circle is at rest, is called the centre. The right lines AC, AD, AF, &c. drawn from the centre to the circumference, are called radii. A diameter is a right line which passes through the centre, and is terminated both ways by the circumference. The line DAL, for example, is a diameter. An

arc is a part of a circumference, as FG. 6. The circumference of a circle is divided into 360 equal parts, called degrees ; each degre

is divided into 60 parts, called minutes, and each minute into 60 parts, called second 7. Two right lines drawn from the same point, and diverging from each other. form a

opening which is called an angle. An angle is commonly expressed by three letters, and it is usual to place in the middle that letter which marks the point whence the lines diverge; thus, we say the angle BAC or DAF

(fig. 224.), and not the angle ABC or ACB. 8. The magnitude of an angle does not depend on the lines

by which it is formed, but upon their distance from each other. How far soever the lines AB, AC are continued, the angle remains the same. One angle is greater than another when the lines of equal length by which it is formed are more distant. Thus the angle BAL (fig. 223.) is greater than the angli CAB, because the lines AB, AL are more distant from each other or include a greater arc than the lines AC, AB. If the legs of a pair of compasses be a little separated an angle is formed; if they be opened wider, the angle becomes greater; if they by brought nearer, the angle becomes less.


Fig. 223.



[ocr errors]



Fig. 224.

Fig. 223.



Fig. 226.

% If the print of a pair of compasses be applied to the point G (fig. 225.), and a cir.

eumference NRB be described, the are NR contained within the two lines GL, GM will measure the magnitude of the angle LGM. If the arc NR, for example, be an

are of 40 degrees, the angle LGM is an angle of 40 degrees. 19. There are three kinds of angles (fig. 226.): a right angle (1), which is an angle of 90

degrees; an obtuse angle(II), which contains more than 90 degrees; and an acute angle

(UI), which contains less than 90 degrees. 11. One line is perpendicular to another when

the two angles it makes with that other line are equal : thus, the line CD (fig. 297.) is perpendicular to the line AB, if

the angles CDA, CDB contain an equal number of degrees. 19. Two lines are parallel when all perpendiculars drawn from one to the other are equal, thus, the lines FG, AB (fig. 228.) are pa

Idddddddd rallel, if all the perpendiculars cd, cd, &c.

are equal. 13. A triangle is a surface enclosed by three right lines, called sides (fig. 229.).

An Equilateral triangle (I) is that which has three sides equal ; an isosceles triangle has only two of its sides equal (II); a sculene

А сссссссс в triangle (111) has its three sides unequal. 14. A quadriluteral figure is a surface enclosed by four right lines, which are called its

sides. 15. A paralldogram is a quadrilateral figure, which has its opposite sides parallel; thus.


Fig. 227.

Fig. 208.

[blocks in formation]


[blocks in formation]
[blocks in formation]



Fig. 231.

Fix. 232.


if the side BC (fig. 230.) is parallel to the side AD, and the side AB to the side

DC, the quadrilateral figure ABCD is called a parallelogram. 16. A rectangle is a quadrilateral figure all the angles

whereof are right angles, as ABCD (fig. 231.). 17. A square is a quadrilateral figure whose sides are

all equal and its angles right angles (fig. 232.). 18. À trapezium is any quadrilateral figure not a

parallelogram. 19. Those figures are equal which enclose an equal space; thus, a circle and a triangle are

equal, if the space included within the circumference of the

circle be equal to that contained in the triangle. 3. Those figures are identical which are equal in all their parts ;

that is, which have all their angles equal and their sides equal, and enclose equal spaces, as BAC, EDG (fig. 233.). It is manifest that two figures are identical which, being placed one upon the other, perfectly coincide, for in that case they must be equal in all their parts. It must be observed, that a line merely so expressed always denotes a right

line. Allom. Two right lines cannot enclose a space; that category requires at least threo



Fig. 233.

[ocr errors]


876. PROPOSITION I. The radii of the same circle are all equal.

The resolution of the line AB about the point A (fig. 234.) being necessary (Defin. 5.) to form the circle BCDFGLB, when in revolving the point B is upon the point C, the whole line AB must be upon the line AC; otherwise two right lines would srekse a space, which is impossible: wherefore the radius AC is equal to the radius AB. In like manner it may be proved that te radi AB, AF, AG, &c. are all equal to AB, and are therelore equal arnong themselves.

$77. Pror. II. On a giren line to describe an equilateral tri

[ocr errors]

Fig. 235

[ocr errors]

Let AB (fig. 235.) be the given line upon which it is required to describe a tria whose three sides shall be equal.

From the point A, with the radius AB, describe the cir. cumference BCD, and from the point B, with the radius BA, describe the circumference ACF; and from the point C, where these two circumferences cut each other, draw the two right lines CA, CB. Then ACB is an equilateral triangle.

For the line AC is equal to the line AB, because these two lines are radii of the same circle BCD; and the line BC is equal to the line AB, because these two lines (Prop. 1.) are radii of the same ci ACF. Wherefore the lines AC and BC, being each equal to the line AB, are equal to another, and all the three sides of the triangle ACB are equal; that is, the triangh equilateral.

878. Prop. III. Triangles which have two sides and the angle subtended or contained them equal are identical.

In the two triangles BAC, FDG (fig. 236.), if the side DF be equal to the side : and the side DG equal to the side AC, and also the angle at D equal to the angle at A, the two triangles are identical.

Suppose the triangle FDG placed upon the triangle BAC in such manner that the side DF fall exactly upon the side equal to it, AB. Since the angle D is equal to the angle A, the side DG must fall upon the side equal to it, AC; also the point F will be found upon the point B, and the point G upon the point C: consequently the line FG must fall wholly upon the line BC, otherwise two right lines would enclose a space, which is im- B possible. Wherefore the three sides of the triangle FDG coincide

Fig. 236 in all points with the three sides of the triangle BAC, and the two triangles have th sides and angles equal, and enclose an equal space; that is (Defin. 20.), they are identical.

879. Pror. IV. In an isosceles triangle the angles at the base are equal.

Let the triangle BAC ( fig. 237.) have its sides AB, AC equal, the angles B and C at the base are also equal. Conceive the angle A to be bisected by the right line AD.

In the triangles BAD, DAC the sides AB, AC are, by supposition, equal; the side AD is common to the two triangles,

Fig. 237. and the angles at A are supposed equal. These two triangles, therefore, have two sides, and the angle contained by them equal. Hence, they are identi (Prop. 3), or have all their parts equal: whence the angles B and C must be equal.

880. Prop. V. Triangles which have their three sides equal are identical.

A In the two triangles ACB, FDG (fig. 238.), let the side AC be equal to the wide FD, the side CB equal to the side DG, and the side AB to the side FG; these two triangles are identical.

Let the two triangles beso joined that the side FG shall coincide with the side A B (fig. 239.), and draw the right line CD. Since in the triangle CAD the side AC is equal to the side AD, the triangle is isoceles; whence (Defin. 13.) the angles m and n at the base are equal.

Since in the triangle CBD the side bĆ is equal to the side BD, the triangle is is sceles; whence (Defin. 13.) the angles r and s at the base are equal.

Because the angle m is equal to the angle n, and the angler equal to the angle s, the whole angle C is equal to the whole angle D.

Lastly, because in the two triangles ACB, A DB the side AC is the side AD and the side CB equal to the side DB, also the angle C equal to the angle D, these two triangles have two sides, and the contained angle equal, and are therefore (Prop. 3.) identical.

881. Prop. VI. To divide a right line into two equal parts.

Let the right line which it is required to divide into two equal parts be AB (fig. 240.). Upon AB draw (Prop. 2.) the equilateral triungle ADB, and on the other side of the same line


[ocr errors]




Fig. 238.

Fig. 239.


Fig. 210.

[ocr errors]
[ocr errors]


Craz. I.

267 AB draw the equilateral triangle AFB, draw also the right line DF; AC is equal w CB.

In the two larger triangles DAF, DBF the sides DA, DB are equal, because they are the sides of an equilateral triangle; the sides AF, BF are equal for the same reason ; and the side DF is common to the two triangles. These two triar.gles, then, have their sides equal, and consequently (Prop. 5.) are identical, or have all their parts equal ; wherefure the two angles at D are equal.

Again, in the two smaller triangles ADC, CDB the side DA is made equal to the ade DB, and the side DC is common to the two triangles; also the two angles at D are equal Thus these two triangles have two sides and the contained angle equal; they are therefore (Prop. 3.) identical, and AC is equal to CB; that is, AB is bisected. 882. Prop. VII. From a given point out of a right line to draw a perpendicular to that

Let C (fig. 241.) be the point from which it is required to draw a perpendicular to the right line AB.

From the point C describe an arc of a circle which shall cut the line AB in two points F and G. Then bisect the line FG, and to D, the point of division, draw the line CD: this line is perpendicular to the line AB. Draw the lines CF, CG.

In the triangles FCD, DCG the sides Cr, CG are equal, betause (Prop. 1.) they are radii of the same circle; the sides FD DG are equal, because FG is bisected; and the side CD is com

Fig. 241. mon. These two triangles, then, having the three sides equal, are identical (Prop. 5.). Whence (Defin. 20.) the angle CDA is equal to the angle CDB, and consequently (Defin. 11.) the line CD is perpendicular to the line AB.

883. Peop. VIII. From a given point in a right line to raise a perpendicular upon that From the point C (fiy. 242.), let it be required to raise a j.erpendicular upon the right line AB.

In AB take at pleasure CF equal to CG; upon the line FG describe an equilateral triangle FDG, and draw the line CD); this line will be perpendicular to AB.

In the triangles FDC, CDG the sides DF, DG are equal, beause they are the sides of an equilateral triangle; the sides FC, A LG are equal by construction; and the side DC is common. These two triangles, then, having the three sides equal, are (Prop. 5.) identical. Therefore (Defin. 20.) the angle DCA is equal to the angle DCB, and bonisequently (Defin. 11.) the line CD is perpendicular to the line AB.

884. Prop. IX. The diameter of a circle divides the circumference into two equal Let ADBLA ( fig. 243.) be a circle; the diameter ACB bisects the circumference, that Is the are ALB is equal to the arc ADB. Conceive the circle to be divided, and the lower segment ACBLA to be placed upon the upper ACBDA; all the points of the are ALB will fall exactly upon the arc ADB; and consepuently these two ares will be equal.

For if the point L, for instance, does not fall upon the arc ADB, & must fall either above this arc, as at G, or below it, as at F. If it fall on G, the radius CL will be greater than the radius CD; ifit falls on F, the radius CL will be less than the radius CD, shieh is (Prop. 1.) impossible. The point L, then, must fall upon Fig. 243. the are ADB. In like manner it may be proved that all the other points of the arc ALB must fall upon the arc ADB: those two arcs are therefore equal.

885. Prop. X. A right line which meets another right line forms with it two angles, which Bagetker, are equal to two right angles.

The line FC (fig. 244.) meeting the line DA, and forming with it the two angles, DCF, ACF, these two angles are together equal to two right angles.

From the point C as a centre describe at pleasure a circumference NGLMN. The line NCL, being a diameter, divides the circumference Prop. 9) into two equal parts. The arc NGL is therefore balf the circumference, which contains (Defin. 6.) 180, or twice 90 degrees. Therefore the angles DCF, ACF, which, taken together, are measured by the arc NGL, are twice 30 degrees, that is (Defin. 10.), are equal to two right angles. BRA. P'ror. XI. A line drawn perpendicularly to another right line makes right ingles

[ocr errors]



Fig. 242.


Fig. 244.

« PreviousContinue »