PAR
Axis, in itsProgrefs thro' its Orbit, whereby it {fill l 00 k stot he Tame Point of the Heavens, viz. towards the Pole Star • fofhat if a Line be drawn parallel to itsAxis, while in any one Pofition ; the Axis, in all other Pofitions or Parts of the Orbit will al- ways be parallel to the fame Line. See Axis. '
This Parallelifm is the neceffary Refult of the Earth '« double Motion ; the one round the Sun, the other round its own Axis. Moris there any Necefflty to imagine a third Motion, as fome have done, to account for this 'Parallelism. See Earth.
'Tis to this Parallelifm that we owe the Viciffitude of Sea- fons, and the Inequality of Day andNight. See Season. See alfo Day, igc.
Parallelism of Rc-zvs of Trees. The Eye placed at the End of an Alley of two Rows of Trees, planted in parallel Lines, never fees 'em parallel, but always inclining to each other, to- wards the further Extreme.
Hence the Mathematicians have taken Occafion to enquire in what Lines, the Trees mull be difpofed, to correct this Ef- fect of the Perfpective, and make the Rows ftill appear paral- lel ; parallel they mutt not be, but diverging ; but according to what Law mull they diverge : The two Rows mult be fuch, as that the unequal Intervals of any two oppofite or correfpond- ing Trees may be feen under equal vifual Angles.
On this Principle, F. Fabry has aflertcd, without any De- monftration, and F. Racquet, after him, demonftrated by along and intricate Synfhefis, that the two Rows of Trees muft be two oppofite Semi-Hyperbola's.
M. Varignon has fince, in the Memoirs of the French Aca- demy, Mm 1 717, found the fame Solution bv an eafy and fimple Analyfis. But he renders the Problem much more gene ral, and requires not only that the vifual Angles be equal, but to have them increafe or decreafe in any given Ratio ; provi- ded the greateft do not exceed a right Angle. The Eye, he requires to be placed in any Point, either juft at the Beginning of the Ranges, or beyond, or on this Side.
All this laid down, he fuppoles the firft Row to be a right Line, and feeks what Line the other muft be, which he calls the Curve of the Range. This he finds mull be an Hyperbola, to have the vifual Angles equal. The ftraight and hyperbo- lical Rows will be feen parallel to Infinity ; and if the oppofite
( 749 ) '
PAR
Semi-hyperbola be added, wefhallhave three Rows of Trees, Triangle
propomonably thereby. Hence, fimilar Parallelograms and Triangles are in a duplicate Ratio of their homologous Sides, as alio of their Altitudes, and the Segments of their Bales : They are, therefore, as the Squares of the Sides,Altitudes, and homologous Segments of the Bafes.
In every Parallelogram, the Sum of the Square!, of the two ■Diagonals, is equal to the Sum of the Square of the four S.des.
This Propofition, M. de Lagny, takes to be one of the moft importantin all Geometry; he even ranks it with the celebrated 47th of Euclid, and with that of the Similitude of Triandes- and adds, that the whole firft Book of Euclid is only a parti- cular Cafe hereof For, if the -Parallelogram be not reef an- gular, it follows that the two Diagonals are equal ; and, of confequence, the Square of a Diagonal, or which comes to the fame Thing, the Square of the Hypothenufe of a right Angle, is equal to the Squares of the two Sides. If the Paral- lelogram be not rectangular, and, of confequence, the two Diagonals be not equal ; which is the moft general Cafe ; the Propofition becomesof yaft Extent. Itmay fare, forlnftance, in the whole Theory of compound Motions, ?£c.
There are three Manners of demonstrating this Problem ; The firft by Trigonometry, which requires 21 Operations; the fecond Geometrical and Analytical; which requires 15. M.de Lagny gives a more concife one, in the Memoirs, de t Aa4, which only requires 7. See Diagonal.
TofiiiitheJreaofarellai7gledVo.rz\\t\ogrmi, A BCD. "
Find the Length of the Sides A B, and A C; multiply AB into A C ; the Produce will be the Area of the Parallelogram. Suppofe E. gr. A B to be 345 ; AC 123. The Area will be 4243;.
Hence 1. Rectangles are in a Ratio compounded of their Sides A B and A C. 2. If, therefore, there be three Lines con- tinually proportional ; the Square of the middle one is equal to the Reftangle of the two Extremes: and if there be four pro- portional Lines; the Reflangle under the two Extremes is equal to that under the two middle Terms. See Rect- angle.
Other Parallelograms, not reflangular, have their Areas found by refolving them, by Diagonals, into two Triangles ; and adding the Areas of the feparate Triangles into one Sum. See
(the ftraight one in the Middle) and all three parallel.
Nor is it required this fecond Hyperbola be the Oppofite of the firll, i. e. of the fame Kind, or have the fame traufverfe Axis : 'Tis enough if it have the fame Centre, its Verrex in the fame right Line, and the fame conjugate Axis. Thus the two Hyperbola's may be of all the different Kinds poflible ; yet all have the fame Effort.
Again, the ftraight Row being laid down as before; if it be required to have the Trees appear under decreafing Angles; M. Varignon fhews, that if the Decreafe be in a certa'in Ratio, which hedetermines; the other Line muft be a parallel ftraioht Line. But he goes yet farther ; and fuppofing the firft R°ow any Curve whatever, he fecks for another that '(hall make the Rows have any Effect defired, /. e. be feen under any Angles equal, increasing, or decreafing.
PARALLELOGRAM, in Geometry, a Quadrilateral Fi-
Centre cf Gravity of a Parallelogram. See Centre of Gravity. See alfo Centrobaryc Method.
fjKllLlloCKH, or Parallelism, or Paralielogramic ProtraBor, is a Machine ufed for the ready and exafl Redufli- on or Copying of Defigns, Schemes, Prints, £Jc. in any Propor- tion ; which is done hereby without any any Knowledge or Habit of Defigning.
The Parallelogram is alfo called Pentagraph. See its De- fcription and Ufe under the Article Pentagraph.
PARALOGISM, in Logic, a falfe Reafoning ; ora Fault committed in a Demonftration, when a Confequence is drawn from Principles that are falfe, or not proved ; or when a Pro- pofition is pafs'd over, which (hou'd have been proved by the Way.
A Paralcgifm differs from a Sofhifm, in this, that the So- phifm is made out of Defign and Subtlety; and the Paralo-
a Difeafe popularly call'd
gure, whole oppofite Sides are parallel, and confequently gifm out of Miftake, and for Want of a fufficient Light and
Application. See Sophism.
Yet the Meflieurs de Port-Royal don't feem to make any Difference between them. None of the Pretenders to the Qua- drature of the Circle but have made Paralcgifms. See Qua- drature.
PARALYSIS, in Medicine, palfy. See Palsy.
The Paralyfu only differs from the Parefis as the greater fromthelefs. See Paresis.
Authors cUftinguiih the Paraly/is into a Panplegia, or pa- raplexia, Hemiplegia, and, particular Paraly/is.
The firft is a Palfy of the whole Body. See Paraplegia. Thefecond, of one Side of the Ecdy. See Hemiplegia. The third of fome particular Member, which is the proper 'Palfy. l r
The Word Paraly/is is form'd from, the Greek irx&ihja, I unbind ; This Difeafe being fuppofed to unbend the Nerves and Mufcles. Hence
PARALYTIC, a Perion affected with the Paraly/is or Palfy. See Palsy, (go. J
PARAMETER, in Geometry, a conftant right Line, in feveral of the Conic Seftions ; call'd alfo Zatus reihim. See Latus Rectum.
In a Parabola VB V Tab. Celtics Fig. 9. the Rectangle of the Parameter A B, and any Abfciffe, E.gr. B 2, and Semi- ordinate 3 III. See Parabola.
If all the Sides, and Angles of a Quadrilateral Figure be e- qual, it is called a Square ; which fome make a Species of Parallelogram, others not. See Squar e.
In an Ellipfis and Hyperbola, the Parameter is a third Pro- portional to to the conjugate and tranfverfe Axis. See Ellip- sis, ££c. Hyperbola.
PARAMOUNT, in our Law, fignifies the fupreme Lord of the Fee. See Lord and Fee.
S E Ther»
equal to each other. See Quadrilateral.
A Parallelogram is generated by the equable Motion of a right Line always parallel to itfelf. See Figure.
When the Parallelogram has all its four Angles right, and only its oppofite Sides equal, it is call'd a Refiangle or oblong. See Rectangle.
When the Angles are all right, and the Sides equal, it is called a Square. See Square. If all the Sides be equal, and the Angles unequal, it is call'd a Rhombus or Lozange. See Rhombus.
If both the Sides, and Angles be unequal, it is call'd a Rhomboides. See Rhomboides.
Properties of the Parallllogram.
In every Parallelogram, what Kind foever it be of, E. gr. that A BCD Tab.Geometry Fig. 39. A Diagonal DA divides it into two equal Parts ; the Angles diagonally oppofite BC, and A D, are equal, the oppofite Angles of the fame Side C D, and AB, '(fiC are, together, equal to two right Angles; and each two Sides, together, greater than the Diagonal.
Two parallelograms A B C D, and E C D F on the fame or equal Bafe C D, and of the fame Height A C, or between the fame Parallels AF, CH are equal. Hence two Triangles C D A and C D F on the fame Bafe, and of the fame Height, are alfo equal. Hence, alfo, every Triangle CFDishalf a Parallelogram A C D B, upon the fame or an equal Bafe C D, and of the fame Altitude, or between the fame Parallels. Hence alfo a Triangle is equal to a Parallelogram, having the fame Bafe, and half the Altitude, or half the Bafe and the fame Altitude. See Triangle.
Parallelograms, therefore, are in a given Ratio, compounded of their Bales and Altitudes. If then the Altitudes be equal, they are as the Bafes, and converfly.
In fimilar Parallelograms and Triangles, the Altitudes are proportional to the homologous Sides ; and the Bafes are cut
