A Ready Description and Quadrature of a Curve of the Third Order, Resembling That Commonly Call'd the Foliate. Communicated by Mr. Abr. De Moivre, F. R. S.

the Vessel; and in a natural State they shut that Passage, and so prevent the Blood from recoiling into the same, if it should endeavour to return. But in this case, by reason of its contracted Narrowness and Thickness, not being able to close or shut the Passage, the Blood flowed back again into the Cavity, which it had gradually enlarged, and dilated to the Bigness we see. Besides the Muscular Valves not being duly qualified for the Performance of their Office, the Blood recoiled into the Auricle, which it had distended in the like manner. This constant Regurgitation or Reflux of the Blood is besides sufficient of itself, to produce this extraordinary trembling or ωαλμος καρδιας, as the Greeks call it. IV. A ready Description and Quadrature of a Curve of the Third Order, resembling that commonly call'd the Foliate. Communicated by Mr. Abr. de Moivre, F. R. S. I have look'd a little farther into that Curve which fell lately under my consideration. It is not the Foliate as I did at first imagine, but I believe it ought not to make a Species distinct from it. $AEB$ (Fig. 1.) is the Curve I thus describe. Let $AB$ and $BK$ be perpendicular to each other. From the point $A$ draw $AR$ cutting $BK$ in $R$, and make $RE = BR$, the point $E$ belongs to the Curve Draw $BC$ making an Angle of 45 grad. with $AB$, this Line $BC$ touches the Curve in $B$; from the point $E$ draw $ED$ perpendicular to $BC$, and calling $BD, x$; $DE, y$; $AB, a$; and making $\sqrt{8aa} = n$, the Equation belonging to that Curve is $x^3 + xx + xyy + y^3 = nxy$ or $\frac{x^4 - y^4}{x - y} = nxy$. Taking $BG = AB$, and drawing $GP$ perpendicular to $BG$, $PG$ is an Asymptote. In the Foliate F f f 3 the Equation is $x^3 + y^3 = \frac{1}{2} n x y$, in which the two Terms $x x y + x y y$ of the former Equation are wanting; and its Asymptote is distant from $B$ by $\frac{1}{3} B A$. Again draw $E F$ perpendicular to $A B$: let $B F$ be called $z$ and $F E v$; the Equation belonging to the Curve $A E B$ is $v v = \frac{a z z - z^3}{a + z}$. In the Foliate the Equation is $v v = \frac{azz - z^3}{a + 3 z}$. From these two last Equations it seems that these Curves differ no more from one another than the Circle from the Ellipsis. I should be very glad to know your Opinion thereupon. The Quadrature of the Curve here described has something of Simplicity with which I was well pleased. With the Radius $B A$ and Center $B$ describe a Circle $A K G$, let the Square $H P S T$ circumscribe it, so that $H P$ be parallel to $A G$: prolong $F E$ till it meet the Circumference of the Circle in $M$, and through $M$ draw $L M Q$ parallel to $H P$. The Area $B F E$ is equal to the Area $K H L M$, comprehended by $K H$, $H L$, $L M$ and the Arc $K M$. And the Area $B f e$ is equal to the Area $K m L H$ or $K M P Q$. Therefore if $B F$ and $B f$ are equal, the two Areas $B F E$, $B f e$ taken together are equal to the Rectangle $H Q$, and therefore the whole Space comprehended by $B E A X B e T G Z$ (supposing $T$ and $Z$ to be at an infinite Distance) is equal to the circumscribed Square $H S$. N.B. This Quadrature is easily demonstrated from the Equation: for by it $a - z : a - z :: z z : v v$, that is $A F : E F :: M F : F B$, and so $F$ the Fluxion of $A F$ to $L l$ the Fluxion of $M F$. Hence the Areola $E F \phi e$ will be always equal to the Areola $M L l u$, and therefore the Area $A E F$ always equal to the Area $M A L$. Hence it appears that this Curve requires the Quadrature of the Circle to square it; whereas the Foliate is exactly quadrable, the whole Leaf thereof being but one third of the Square of $A B$, which in this is above three sevenths of the same. Again in our Curve, the greatest Breadth is when the Point F divides the Line A B in extreme and mean Proportion: whereas in the Foliate it is when A B is triple in power to B F. And the greatest E F or Ordinate in the Foliate is to that of our Curve nearly as 3 to 4, or exactly as $\sqrt{\frac{2}{3}} \cdot \sqrt{\frac{1}{3} - \frac{1}{3}}$ to $\sqrt{5} \cdot \sqrt{\frac{4}{5} - \frac{4}{5}}$. But still these Differences are not enough to make them two distinct Species, they being both defined by a like Equation, if the Asymptote S G P be taken for the Diameter. And they are both comprehended under the fortieth Kind of the Curves of the third Order, as they stand enumerated by Sir Isaac Newton, in his incomparable Treatise on that Subject. IV. An easy Mechanical Way to divide the Nautical Meridian Line in Mercator's Projection; with an Account of the Relation of the same Meridian Line to the Curva Catenaria. By J. Perks, M. A. The most useful Projection of the Spheric Surface of Earth and Sea for Navigation, is that commonly call'd Mercator's; tho' its true Nature and Construction is said to be first demonstrated by our Countryman Mr. Wright, in his Correction of the Errors in Navigation. In this Projection the Meridians are all parallel Lines, not divided equally, as in the common plain Chart (which is therefore erroneous,) but the Minutes and Degrees (or strictly, the Fluxions of the Meridian,) at every several Latitude are proportional to their respective Secants. Or a Degree in the projected Meridian at any Latitude, is to a Degree of Longitude in the Equator, as the Secant of the same Latitude is to Radius.