PAR 



PAR 



axis, in its progress through its orbit, 

 whereby it is .still directed towards the 

 pole-star ; so that it' a line be drawn pa- 

 rallel to iis axis, while in any one position, 

 the axis, in all other positions, will be al- 

 ways parallel to the same line. 



This parallelism is the result of the 

 earth's double motion, viz. round the sun, 

 and round its own axis ; or its annual and 

 diurnal motion ; and to it \ve owe the vi- 

 cissitudes of seasons, and the inequality 

 of day and night. 



PARALLELISM of the rows of trees. 

 These are never seen parallel, but always 

 inclining to each other towards the fur- 

 ther extreme. Hence mathematicians 

 have taken occasion to inquire in what 

 lines the trees must be disposed to cor- 

 rect this eifect of the perspective, and 

 make the rows still appear parallel. The 

 two rows must be such, as that the une- 

 qual intervals of any two opposite or cor- 

 respondent trees may be seen under equal 

 visual rays. 



PARALLELOGRAM, in geometry, a 

 quadrilateral right-lined figure, whose 

 opposite sides are parallel and equal to 

 each other. It is generated by the equa- 

 ble motion of a right line always parallel 

 to itself. When it has all its four angles 

 right, arid only its opposite sides equal, 

 it is called a rectangle or oblong. When 

 the angles are all right, and the sides 

 equal, it is called a square. If all the 

 sides are equal, and the angles unequal, 

 it is called a rhombus or lo/enge ; and if 

 the sides and angles be unequal, it is call- 

 ed a rhomboides. 



In every parallelogram, of what kind 

 soever, a diagonal divides it into two 

 equal parts ; the angles diagonally oppo- 

 site are equal ; the opposite angles of the 

 same side are together equal to two right 

 angles ; and each two sides, together, 

 greater than the diagonal. 



Two parallelograms on the same or 

 equal base, and of the same height, or 

 between the same parallels, are equal ; 

 and hence two triangles on the same base, 

 and of the same height, are also equal. 

 Hence, also, every triangle is half a pa- 

 rallelogram, upon the same or an equal 

 base, and of the same altitude, or between 

 the same parallels. Hence, also, a trian- 

 gle is equal to a parallelogram, having 

 the same base, and half the altitude, or 

 half the base, and the same altitude. 



Parallelograms, therefore, are in a gi- 

 ven ratio, compounded of their bases and 

 altitudes. If then the altitudes be equal, 

 they are as the bases, and conversely. 



In similar parallelograms and triangles, 



the altitudes are proportional to the ho- 

 mologous sides, and the bases are cut 

 proportionally thereby. Hence similar 

 parallelograms and triangles are in a du- 

 plicate ratio of their homologous sides ; 

 as also of their altitudes, and the seg- 

 ments of their bases ; they are, therefore, 

 as the squares of the bides, altitudes, and 

 homologous segments of the bases. 



In every parallelogram, the sum of the 

 squares of the two diagonals is equal to 

 the sum of the squares of the four sides. 

 For if the parallelogram be rectangular, 

 it follows that the two diagonals arc 

 equal ; and, consequently, the square of 

 a diagonal, or, which comes to the same 

 tiling, the square of the hypothenuse of 

 a right angle, is equal to the squares of 

 the sides. See GEOMETRY. 



PARALLELOGRAM, or PARALLELISM, 

 a machine for the ready reduction of de 

 signs ; it is the same with the PENT A 

 GRAPH, which see. 



PARAMETER, in conic sections, a 

 constant line, otherwise called latus rec- 

 tum. The parameter is said to be con- 

 stant, because, in the parabola, the rec- 

 tangle under it and any absciss, is always 

 equal to the square of the corresponding 

 semi-ordinate ; and in the ellipsis and hy- 

 perbola, it is a third proportional to the 

 conjugate and transverse axis. 



If t and c be the two axes in the ellipse 

 and hyperbola, and x and y an absciss 

 and its ordinate in the parabola : then 



t .- c :.-c : p = = the parameter in the 



former ; 

 x : y :: y:p=~-= the parameter in 



the last. 



The parameter is equal to the double 

 ordinate drawn through the focus of one 

 of the three conic sections. 



PAUAMEC1UM, in natural history, a 

 genus of the Vermes Infusoria class and 

 order. Worm invisible to the naked eye, 

 simple, pellucid, flattened, oblong. There 

 are seven species, of which P. aurelia is 

 rather a large animalculum, ruembrana- 

 ceous, pellucid, and about four times 

 longer than it is broad ; the fore-part ob- 

 tusf, transparent, without intestines ; the 

 hind-part replete with molecules of va- 

 rious sizes ; the fold which goes from 

 the middle to the apex is a striking char- 

 acteristic of the species, forming a kind 

 of triangular aperture, and giving it 

 somewhat the appearance of a gimhlet. 

 Its motion is rectilinear, reeling or stag- 

 gering, and generally vehement. They 

 are frequently found cohering lengthwise ; 





