Preface

...they will hasten toward the center by the shortest shortcut. However, we have discussed very little regarding oblique motion, which occurs on an inclined plane. For that reason, some points must be taken from Propositions 7 and 8 of Book 2 of the French work Mersenne refers to his own Harmonie Universelle (1636), written in French.. Therefore, in the figure CAB, let CA be the oblique plane, AB the horizontal plane, and CB the vertical line.

A geometric diagram shows a right-angled triangle ABC. The hypotenuse AC represents an inclined plane. AB is the horizontal base and BC is the vertical side. A line extends upwards from C to a point T, and a line is drawn from B to T. Points D and I are marked on or below the inclined plane, with lines indicating paths of motion.

With these things established, Galileo concludes that a heavy body falls from C to B and from C to T in the same amount of time. Thus, the point on the inclined plane reached by falling bodies is found by means of a perpendicular line drawn to the plane AB from point B, which defines the vertical motion, that is, by means of the line BT. Similarly, a weight will fall from D to I on the plane DA in the same amount of time it would fall from D to B. Furthermore, one can find the location on the vertical line CB, extended further toward the center, which the heavy body reaches when it descends from C all the way to A. This is found if a perpendicular is drawn to point A, or if a line drawn to A is made parallel to line IB. It is clear from the same figure that the times for both types of fall are proportional to the lengths of the planes themselves. That is, the time for the vertical fall CB is to the time for the oblique fall CA as the length BC is to CA. Therefore, in the following figure, a weight will fall from A to D in the same time it falls from A to B. So, if AC is assumed to be

A small geometric diagram on the left shows an inclined plane starting from point A. Points B, C, D, and E are marked with intersecting lines and arcs. These illustrate the geometry of motion and angles relative to a central point, likely the center of the earth.

three feet, a heavy body would fall from A to C in a space of 2/3 of a second original: "2''". In this period, double primes often designated thirds, which are 1/60 of a second, or specific fractional parts of a unit., and from C to D in the space of another half second. Therefore, it falls from A to B in the space of one second original: "unius secundi minuti." This refers to a second of time.. From this, it follows that the time of descent along AB is double the time it takes to descend from A to C, just as the distance BA is double AC. In short, the time of oblique descent exceeds the time of vertical fall in the same ratio as the length of the oblique plane exceeds the vertical plane, provided both weights falling from the same point approach the center equally.

See the cited proposition of the French book. There, it is demonstrated that heavy bodies descend more slowly as the plane becomes more inclined. This means the closer the plane gets to the horizon, which you might call infinite inclination original: "infinita inclinationis." This describes a horizontal plane where the time required for a weight to "fall" across it would be infinite because there is no vertical drop.. Thus, it is possible for a falling weight not to cover even a foot of distance in an entire century of one hundred years. Furthermore, whatever logic might seem to dictate in this matter, we have learned through very accurate experiments repeated many times that this is not entirely true. I report these findings very faithfully in that same place. For example, on a plane inclined at fifty degrees, a body would fall 3 and 1/2 feet in the time it takes to fall five feet vertically. However, it actually only covers 2 and 1/4 feet. A surprising thing arose in these motions: a sphere descends faster on a plane inclined at 45 degrees (covering 3 and 1/3 feet) than on a plane of 50 degrees, even though 50 degrees is just as far from 45 as 40 is. Therefore, let Galileo see to this and test it before he makes a proclamation.

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PROPOSITION IV.

To describe a plane of the same inclination, extending from the circumference of the earth to the center, upon which heavy bodies placed at any point would always maintain the same weight balance.

It is certain that straight inclined planes have as many different inclinations relative to the horizon as they have points. We demonstrate this at length in the eighth proposition of the aforementioned French book. This is evident from the straight line EF touching the surface of the earth at point D, where it is solely horizontal. At other points like 10, 20, 30, and so on up to F, it makes different angles with the horizontal line DO of point D. For example, let the line 10-O fall from point 10 to the center of the earth O. I say that point 10 on the line EF is inclined at 10 degrees above the horizon or the horizontal point D. Since the angle OD-10 is a right angle, angle O-10-D will be the complement of angle 10-OD, which is 80 degrees.

Therefore, the further individual points on the line DF move from point D, the more they will be inclined. The angle at the center of the earth always increases. Hence, heavy bodies will fall faster until they reach point D. We have discussed at great length in the cited passage how much more or less heavy bodies press upon any point of line DF. For example, on a point of that line inclined at 45 degrees, the weight alone will press the plane with its parts.

But a plane of constant inclination will be described if it cuts all lines drawn from the center at equal angles. This occurs in this figure with lines D-α, β, γ, etc., or D-a-b-c-d, etc., or D-χ, λ, μ, etc., and D-1, 2, 3, 4, etc., which approaches closest to the center. If the circumference D, 350, 340, etc., divided into 360 parts, represents the Equator, and the lines drawn from the center represent Meridians and all concentric circles parallel to the Equator, then the aforementioned planes of uniform inclination everywhere represent the path of a ship Mersenne is describing a loxodrome or rhumb line, a curve that crosses all meridians at the same angle.. This path cuts all Meridians at the same angles and will never reach the pole. You will find the very difficult calculation in the cited location, since...