PAR 



PAR 



gel, 8cc. are disseminated far and wide. 

 In some plants, as hawk-weed, the pap- 

 pus adheres immediately to the seed ; in 

 others, as lettuce, it is elevated upon a 

 foot-stalk, which connects it with the 

 seeds. In the first case it is called pappus 

 sessilis ; in the second, pappus stipita- 

 tus : the foot-stalk, or thread, upon which 

 it is raised is termed " stipes." 



PAR, in commerce, signifies any two 

 things equal in value ; and in money af- 

 fairs, it is so much as a person must give 

 of one kind of specie to render it just 

 equivalent to a certain quantity of ano- 

 ther. In the exchange of money with 

 foreign countries, the person to whom a 

 bill is payable is supposed to receive the 

 same value as was paid the drawer by 

 the remitter ; but this is not always the 

 case, with respect to the intrinsic value 

 of the coins of different countries, which 

 is owing to the fluctuation in the prices of 

 exchange amongst the several European 

 countries, and the great trading cities. 

 The par, therefore, differs from the course 

 of exchange in this, that the par of ex- 

 change shews what other nations should 

 allow in exchange, which is rendered 

 certain and fixed by the intrinsic value of 

 the several species to be exchanged : 

 but the course shews what they will allow 

 in exchange ; which is uncertain and 

 contingent, sometimes more, and some- 

 times less ; and hence the exchange is 

 sometimes above, and sometimes under 

 par. See EXCHANGE. 



PARABOLA, in geometry, a figure 

 arising from the section of a cone, when 

 cut by a plane parallel to one of its sides. 

 See CONIC SECTIONS. 



To describe a parabola in piano, draw 

 a right line A B (Plate Parabola, fig. 1) 

 and assume a point C without it ; then, in 

 the same plane with this line and point, 

 place a square rule D E F, so that the 

 side D E may be applied to the right line 

 A B, and the other E F turned to the side 

 on which the point C is situated. This 

 done, and the thread F G C, exactly of 

 the length of the side of the rule, E F, 

 being fixed at one end to the extremity 

 of the rule F, and at the other to the 

 point C, if you slide the side of the rule, 

 D E, along the right line A B, and by 

 means of a pin, G, continually apply the 

 thread to the side of the rule, E F, so as 

 to keep it always stretched as the rule is 

 moved along, the point of this pin will 

 describe a parabola G H O. 



Definitions. 1. The right line A B is 

 called the directrix. 2. The point C is 

 the focus of the parabola. 3. All per- 



pendiculars to the directrix, as L K, M 0, 

 &c. are called diameters ; the points, 

 where these cut the parabola, are called 

 its vertices ; the diameter B I, which pas- 

 ses through the focus C, is called the 

 axis of the parabola ; and its vertex, IF, 

 the principal vertex. 4. A right line, 

 terminated on each side by the parabola, 

 and bisected by a diameter, is called the 

 ordinate applicate, or simply the ordinate, 

 to that diameter. 5. A line equal to four 

 times the segment of any diameter, in- 

 tercepted between the directrix and the 

 vertex where it cuts the parabola, is call- 

 ed the latus rectum, or parameter of 

 that diameter. 6. A right line which 

 touches the parabola only in one point, 

 and being produced on each side falls 

 without it, is a tangent to it in that point. 



Prop. 1. Any right line, as G E, drawn, 

 from any point of the parabola, G, per- 

 pendicular to A B, is equal to a line, G C, 

 drawn from the same point to the focus. 

 This is evident from the description ; for 

 the length of the thread, F G C, being 

 equal to the side of the rule E F, if the 

 part F G, common to both, be taken away, 

 there remains E G = G C. Q. E. D. 



The reverse of this proposition is 

 equally evident, viz. that if the distance 

 of any point from the focus of a para- 

 bola be equal to the perpendicular 

 drawn from it to the directrix, then shall 

 that point fall in the curve of the parabola. 



Prop. 2. If from a point of the parabola, 

 D, (fig. 2) a right line be drawn to the 

 focus, C ; and another D A, perpendicular 

 to the directrix ; then shall the right line, 

 D E, which bisects the angle, A D C, con- 

 tained between them, be a tangent to the 

 parabola in the point D : a line also, as 

 H K, drawn through the vertex of the 

 axis, and perpendicular to it, is a tangent 

 to the parabola in that point. 



1. Let any point F, be taken in the line 

 D E, and let F A, F C, and A C be joined ; 

 also let F G be drawn perpendicular to 

 the directrix. Then, because (by Prop. 

 1), D A = D C, D F common to both, 

 and the angle F D A = F D C, F C will 

 be equal to F A ; but F A greater than 

 F G, therefore F C greater than F G, and 

 consequently the point, F, falls without 

 the parabola : and as the same can be de- 

 monstrated of every other point of D E, 

 except D, it follows that D E is a tangent 

 to the parabola in D. Q. E. D. 



2. If every point of H K, except H, 

 falls without the parabola, then is H K a 

 tangent in H. To demonstrate this, from 

 any point K, draw K L perpendicular to 

 A B, and join K C ; then because K C is 



