As to infinite Surface or Area, any right Line, infinitely extended both ways on an infinite Plane, does divide that infinite Plane into equal Parts; the one to the right, and the other to the left of the said Line: But if from any Point in such a Plane, two right Lines be infinitely extended, so as to make an Angle, the infinite Area, intercepted between those infinite right Lines, is to the whole infinite Plane, as the Arch of a Circle, on the Point of Concourse of those Lines, as a Centre, intercepted between the said Lines, is to the Circumference of the Circle; or as the Degrees of the Angle to the 360 Degrees of a Circle. For Example, two right Lines meeting at a right Angle do include, on an infinite Plane, a quarter part of the whole infinite Area of such a Plane.
But if so be, two parallel infinite Lines be supposed drawn on such an infinite Plane, the Area intercepted between them will be likewise infinite; but at the same time will be infinitely less, than that Space which is intercepted between two infinite Lines that are inclined, though with never so small an Angle; for that in the one Case, the given finite distance of the parallel Lines diminishes the Infinity in one Degree of Dimension; whereas in a Sector, there is Infinity in both Dimensions; and consequently, the Quantities are the one infinitely greater than the other, and there is no proportion between them.
From the same Consideration arise the Three several Species of infinite Space or Solidity, as has been said; for a Parallelepipede, or a Cylinder, infinitely long, is greater than any finite Magnitude how great soever; and all such Solids, supposed to be formed on given Bases, are as those Bases, in proportion to one another. But if two of these three Dimensions are wanting, as in the Space intercepted between two parallel Planes infinitely extended, and at a finite distance; or with infinite Length and Breadth, with a finite Thickness; All such Solids shall be as the given finite distances one to another: But these Quantities, though infinitely greater than the other, are yet infinitely less than any of those, wherein all the three Dimensions are infinite. Such are the Spaces intercepted between two inclined Planes infinitely extended; the Space intercepted by the Surface of a Cone, or the sides of a Pyramid likewise infinitely continued, &c. of all which notwithstanding, the Proportions one to another, and to the τὸ πᾶν, or vast Abyss of infinite Space (wherein is the Locus of all things that are or can be; or to the Solid of infinite Length, Breadth, and Thickness, taken all manner of ways) are easily assignable. For the Space between two Planes, is to the whole, as the Angle of those Planes to the 360 Degrees of the Circle. As for Cones and Pyramids, they are as the Spherical Surface, intercepted by them, is to the Surface of the Sphere; and therefore Cones are as the versed Sines of half their Angles, to the Diameter of the Circle: These three sorts of infinite Quantity are analogous to a Line, Surface and Solid, and after the same manner cannot be compared, or have any proportion the one to the other.
Besides these, there are several other Species of infinite Quantity, arising from the Contemplation of Curves, and their Asymptotes; which, by reason of the difficulty of the Subject, cannot be made so plain to most Readers: But what has been already said, may be sufficient to evince what we undertook to explain.
An Account of Dr. Robert Hook's Invention of the Marine Barometer, with its Description and Uses; published by order of the Royal Society, by E. Halley, R. S. S.
SInce it was found that the Torricellian Tube, commonly call'd the Mercurial Barometer, by the rising and falling of the Quick-silver therein, doth presage the Changes of the Air, in relation to fair and foul Weather; upon several Years Observation of it, it has been proved and adjusted for that purpose by Dr. Robert Hook; and there have been by him many attempts to improve the Instrument, and render the Minute Divisions on the Scale thereof more sensible. He also judging that it might be of great use at Sea, contrived several ways to make it serviceable on Board of Ship; one of which he explain'd to the Royal Society at their Weekly Meeting in Gresham College, January 2. 1667/8. Since which time he hath further cultivated the Invention, and some Years ago produced before the said Society, the Instrument I am now to describe, which for its subtilty and usefulness, seemeth to surpass all other performances of the like Nature.
'Till such time as the Author's present Indisposition will give him leave to bestow freely his Thoughts on this Subject upon the Publick, it is the Opinion of the Society, that such an Account be given of this Contrivance, as may render it known, and recommend it to the Mariners use, for which it was principally intended.
The Mercurial Barometer requiring a perpendicular Posture, and the Quick-silver vibrating therein with great Violence upon any Agitation, is therefore uncapable of being used at Sea (tho' it hath lately been contrived to be made portable), so it remain'd to find out some other Principle, wherein the Position of the Instrument was not so indispensably necessary: For this, all those that use the Sea are obliged to the great facility Dr. Hook has always shewn, in applying Philosophical Experiments to their proper uses.
It is about forty Years since, that the Thermometers of Robert de Fluctibus, depending on the Dilatation and Contraction of included Air by Heat and Cold, have been disused, upon discovery that the Airs pressure is unequal; that inequality mixing it self with the Effects of the warmth of the Air in that Instrument. And instead thereof was substituted the seal'd Thermometer, including Spirit of Wine (first brought into England, out of Italy, by Sir Robert Southwell) as a proper Standard of the temper of the Air, in relation to Heat and Cold; that Ætherial Spirit being of all the known Liquors the most susceptible of Dilatation and Contraction, especially with a moderate degree of either Heat or Cold. Now this being allow'd as a Standard, and the other Thermometer that includes Air, being graduated with the same Divisions, so as at the time when the Air was included, to agree with the Spirit-Thermometer in all the degrees of Heat and Cold, noting at the same time the precise height of the Mercury in the common Barometers: It will readily be understood, that whensoever these two Thermometers shall agree, the pressure of the Air is the same it was, when the Air was included, and the Instrument graduated: That if in the Air-Thermometer the Liquor stand higher than the Division marked thereon, corresponding with that on the Spirit-glass, it is an indication that there is a greater pressure of the Air at that time, than when the Instrument was graduated. And the contrary is to be concluded, when the Air-glass stands lower than the Spirit, viz. that then the Air is so much lighter, and the Quick silver, in the ordinary Barometer lower than at the said time of Graduation.