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III. THE LOGARITHMIC CURVE.
68. If after having taken any point A in an indefinite right line GH, we raise ordinates PM which have for logarithms their abscissæ AP, the curve BMm,
M. which passes through the extremities of these ordinates is called the logarithmic curve, Call AP = *, PM =y, m
B =the modulus, e=the number 2.7182818, &c. whose hyper. O bolic logarithm is unity, and we shall have x=m log y=x
loge, HQ AT E PPG or y*=e, whence y=", the equation of the logarithmic curve.
This shews 19, that this curve is of the number of transcendental curves ;
2ly. That when x=0, y or AB=1; 3ly. That if :=AE=AB=1,
y or EF=e
and therefore that making EF=a we shall always have y=a"; hence if the absciesą form the arithmetical progression 1, 2, 3, 4, &c. the ordinates will form the geometrical progression a', a", a', a*, &c. Consequently the logarithmic extends indefinitely above AP. But if on AQ we take negative abscissæ x=- 1, x = -2,
1 1 1 -3, &c. the ordinates will successively become
a' a'2 &c. ; that is, the curve has an infinite branch BO which approaches nearer and nearer to the directrix, or axis GH, without ever touching it.
69. The most remarkable property of the logarithmic, is that its subtangent is always of the same magnitude. This is proved with the greatest facility by the fluxional or differential calculus; but not to anticipate matters, we shall here give a demonstration on nearly similar principles.
Draw the ordinate mp infinitely near MP, and prolong the side Mm to have the tangent MT. Then, if we draw Mr parallel to the axis, and call Pp = e, mr=i, we shall have xte=Ā. log (y +i)
i3 y (1
+ -&c.); there y
Rya By fore since =A log y, we must have
Syo But the quantity i being infinitely little, its powers :, :, &c. may be rejected; we have therefore it or PT=A. Hence the sub
tangent is always equal to the modulus ; and since in general x=A log y, it is clear that in different logarithmic curves, the abscisse of the same ordinates are as the subtangents, or which amounts to the same thing, the logarithms of the same numbers in different systems are to each other as the moduli.
70. If a circle AG be made to roll along a right line Aa, till the point which first touched this right line at A, again touches it at a, this point will describe a curve called a cycloid, or trochoid. The G E B
F labours of Pascal, Huyghens, Bernoulli, and other eminent mathematicians have rendered this curved very celebrated.
The curve will be a common cycloid when the generating circle has no other movement but that of its revolution.
B But if it has also a movement of translation
P in the same direction the point A will describe curtate cycloid.
A If this movement be in a contrary
B direction the curve is called a prolate or lengthened cycloid.
M It is clear that in the common cycloid, fig. 1. the base Aa is equal to the circumference of the generating circle ;
A that it is shorter in the curtate cycloid, and that it is longer in the prolate cycloid.
The diameter BC of the generating circle is called the axis of the cycloid when it is perpendicular to the middle of the base. The point B is the vertex, and therefore BC is its greatest height.
71. This premised, draw MP perpendicular on BC, (fig. 1) and draw the equal chords MF and OC, we shall have FC =
MO, because of the parallel lines ; therefore since FC-AC-AF=BOC FKM=BOC-OLC=BIO, it is evident that the part MO of the ordinate MP is always equal to the corresponding arc BIO of the generating circle. Besides the other part OP is the sine of that same arc; therefore calling MP=y, BIO=u, we shall have for the equation of the common cycloid y = 4 + sin u, and to render this equation general, make MO = BIO, which agrees to the com
mɔn, curtate or prolate cycloid, according as 6 is equal to, lens, or greater than a : so that the general equation will be y = +sin 4. Hence the cycloid is a transcendental or mechanical curve.
72. To draw a tangent MT to the point M, conceive the arc Mm to be infinitely small, draw
T the ordinate mp, and the little
3 line Mr parallel to the tangent OT at the point of the cir.
P cumference of the generating circle. We shall have MO 6
BIO and mo =
6 gives mr = Oo. Also, be- A cause of the similar triangles, we have
MO X Mr mr : Mr::MO: OT=1
we must take therefore on the tangent of the generating circle the part OT=BIO, and draw through the points M and T, the line MT which will be a tangent to the cycloid, whether common, curtate, or prolate. However in the first, the construction may be simplified : for since MO = BIO = OT, we have the angle TOP, or 2BOP = 2TMO; that is, a right line MT parallel to the chord OB is necessarily a tangent to the point M of the common cycloid.
73. Now let there be drawn the indefinite line BQQ' perpendicular on the axis BC, and Q2, Q'm, parallel to the same axis; because of the similar triangles, we shall have mq : Mą, or Q'Q: Pp ::OP: BP; therefore Q'Q * BP-Pp X OP, or MmQQ = Ppoo ; and consequently the circular space BIOP = BQM, and the semicircle BOČB – BDAB. But the rectangle AB in the common cy: cloid is quadruple this semicircle, therefore the cycloidal area is triple the generating circle.
74. If instead of taking a point of the circumference of the circle to describe the cycloid, we had taken a point either within or without the circle, then the curve described would have been another species of cycloid; and if in place of causing the circle to revolve along a right line, we had made it revolve upon
the circumference of another circle, the curve described by one of its points would have been of the species of curves called cpicycloids.
V. THE QUADRATRIX OF DINOSTRATES.
75. Suppose that a straight line AG, tangent at A moves uni. formly parallel to itself along the diameter Aa, and that at the same instant of its setting out from the point A, the radius AC turns uniformly about the centre C towards the point E, so that it coincides with CE at the moment in which the right line AG also coincides with it; by the continual intersection of these two lines we shall have a curve AMD, called the quadratrix.
From this description it follows that any space AP passed over by the right line AG is to the circular arc AB described in the same
time by the extremity of the radius, as any other space AC passed over by this right line is to the corresponding arc ABE, described by the radius. Call AP=1, PM=y, AP=U, AC=a, ABE=90o=c, we shall have Io. m:a::4:0:: angle ACB : angle ACE, therefore
.. 2ly. CP : PM::CA: AG, or a-*:y::a: tang u, there
cx fore y = tang ; and this will be the equation of the co-ordinates of the quadratrix when the point A is the origin of the abcissæ.
76. But if we place their origin at the centre C, making CP=x, we shall have y=
c? x? it (a
&c. 2. 3. 4. a?
2a3 2. 3. 4. a?
-&c. 2. 3. as 2. 3. 4. 5. a
2.3. as . 22.214.171.124.ao therefore when x = 0, y which will become the base CD, will have
for its expression .; consequently if the base of the quadratrix
were known, we should immediately have the quadrature of the cir.
cle, it is this circumstance which gave the name of quadratrix to this curve.
77. If from the centre C and with the radius CD we describe the quadrant DLK, its length will be equal to the radius CA ; for
: DLK::a: c, therefore DLK=a. We shall also have PC=the
arc LD; for : KL::a:u; therefore KL=&=AP, and PC= LD.
78. Let us now take negative abscissæ AP, and substitute their value in the first equation. It will become y=
tang which gives the negative ordinates PM. Therefore the curve has a branch AM', to which we shall find that a right line QN drawn at the distance AQ = a is the asymptote, by supposing y infinite ; for then we shall find that tang = 0, and consequently that x=a.
If after having coincided with CE, the right line AG, and the radius CA continue to move, the one descending towards a, the other revolving in the same direction ; it is visible that their intersection will describe the part Da of the quadratrix.
It is likewise evident that if this curve could be described geome. trically we should immediately be able to assign an angle of any number of degrees, for example of 290°. For this purpose we need only divide AC at the point P, so that AP should be to AC :: 1:m; for then drawing the ordinate PM, and the radius CB, the
1 angle ACB would = 90° since x : a :: 4:0::1:m.
VI. THE SPIRAL OF ARCHIMEDES.
This name is given to the curve CKMA, described by a point C, moving uniformly along the radins CA, during the
M uniform revolution of this radius about
В. the centre C; so that when the radius has passed through the entire circumference this point may coincide with the point
If after having prolonged the radius CA, we cause it to make a