This chapter contains sections 190 to 213
Section 190 edit
To measure the stars of the heavens, in particular their degrees above the horizon at any hour, take for yourself a board and etch thereon a quarter circumference of a circle. Then take a piece of wood that is half the diameter of this circle and affix it by a peg to the center of the circle in such a way as the left end [WHY LEFT? ISN’T IT THE RIGHT SIDE?] of the wooden piece points toward the center. Affix it in such a way that it may be rotated to any point that you wish. Then attach to the left side of the wooden pointer two small pieces of wood. A hole should be made in each of these wood pieces so that one can see straight through both holes at once [in a line-of-sight]. Finally, along the circumference inscribe ninety degrees from point one to two, like so:
Section 191 edit
If one then wishes to know the position of the sun, moon, or stars by degrees over the horizon, in any of the four directions of the world, wherever one may be, take this tool and stand in upon a flat surface that does not incline in any direction so that side CB is towards the ground and side AC is at a right angle over the plane of the earth. Then move the pointer, notated in the figure above as DC, until the desired star is seen through the holes. The left side of the pointer will then show at what point on the circumference is the star. For example, if the star is seen at point E, count the number of degrees from point B to point E. That amount is the number of degrees this star is above the horizon, because all circles, whether large or small, have the same system of degrees. Therefore, the degrees shown on the tool are the same as the degrees of the sphere of the sky right up to point A, which is directly overhead.
Section 192 edit
To measure the height in degrees of the 'pivot of the world' [celestial pole], which is the point in the sky that does not move at all, that rotates around itself - if one knows the star that is in this place one can measure as was mentioned in the previous section. If one cannot locate this star there are several alternative methods.
Section 193 edit
The first method: If you know the number of degrees that the sun is removed from the 'equalizing line' on that very day [this is the line encircling the globe under which the day and night are split equally in both the summer and winter, also known as the ecliptic], - which can be determined by knowing how many days it is from the beginning of the season - and then measure the height of the sun over the celestial horizon [the circle formed by the horizon], the sun's appearance within the southern constellations of Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces means that the distance in degrees between the sun and the ecliptic should be added to the height of the sun over the celestial horizon and that is the height of the ecliptic over the celestial horizon. The remaining degree measure between the ecliptic and 90 degrees is the height of the 'pivot', as is known.
If the sun is found in the northern constellations, which are Aries, Taurus, Gemini, Cancer, Leo and Virgo, subtract the distance of the sun from the ecliptic from its height over the celestial horizon and that equals the height of the ecliptic over the celestial horizon. The degree measure between that height of the ecliptic and 90 degrees is height of the pivot [the celestial pole].
If the sun is at the beginning of Aries or Scales [meaning it is found at the vernal or autumnal equinox, the points where the ecliptic and the celestial horizon meet] then the height of the sun is the height of the ecliptic.
Section 194 edit
Method two: If through some calculation the distance between a star [or planet] and the celestial horizon is already known, whether to the north or south, the measurement of that star from the ecliptic together with the calculation in the previous section will yield the height of the pivot [or celestial pole]. Similarly, if the distance between a star and the pivot is known - if it is on the southern side of the pivot its distance from the pivot is subtracted from its height over the northern celestial horizon. If it is to the north of the pivot [celestial pole], add its distance from the pivot to its height over the celestial horizon and that is the height of the pivot [pole].
Section 195 edit
Method three: Using the astrolabe [constructed in section 190], position it along the 'half-day horizon' [the celestial meridian], a circle running from the north up to the zenith, which is the top of the celestial sphere, and then down to the south. This line passes through the celestial pole.] Finding a star along this line south of the celestial pole, measure its height above the northern celestial horizon. Afterward, wait twelve hours until that star returns to a position along the celestial meridian, this time north of the celestial pole [as all stars in the sky rotate around the celestial pole once in twenty four hours]. Then measure its height over the celestial horizon. When the second height measurement is subtracted from the first, and the remainder is divided in half and either added to the second measurement or subtracted from the first, and the result is the height of the celestial pole over the celestial horizon.
Section 196 edit
To determine the diameter of the sun [in degrees] measure the height over the celestial horizon of the top of the sun [on the celestial pole side] and measure the height over the celestial horizon of bottom edge of the sun [on the celestial horizon side]. The height of the top less the bottom is the [degree] measurement of the sun's diameter
Section 197 edit
Another method is to measure the length of time it takes from the very start of the sun's rising over the land [in the morning] until its rising is complete. The proportion of the time that took to 24 hours [a whole day] will be the same proportion of 360 degrees to the degree measure of the sun's diameter.
The same would apply to the moon and other stars [planets] as well.
Section 198 edit
In order to determine how many 'parsaos' [a land measurement, similar to miles, roughly 2.4–2.88 miles] the diameter or circumference of the Earth is, measure a star that is along the celestial meridian, calculating the distance of that star [or planet] from the zenith [overhead point] or from the celestial horizon. Following the celestial meridian along a direct north or south line, travel to another location and measure this star again. Subtract one measurement from the other. The remainder will be a portion of the degrees of the Earth’s circumference. Calculate the distance between the two points and set up the following proportion: The ratio of the [remainder] amount of degrees to a full 360 degrees is the same as the ratio of the distance between the two points and the Earth’s circumference. Once the circumference is known the diameter can be calculated based on the ratio of 22 to 7 [Pi, 3.14], as was written in section 77.
Section 199 edit
Another method is to find a location where the sun is directly overhead at midday and then measure how many for ‘parsaos’ [north-south or east-west] there are no shadows cast. Afterward measure the diameter of the sun in degrees. The ratio of the diameter of the sun to 360 degrees is proportional to the ratio of the distance measured to the Earth’s circumference. Then the Earth’s diameter can also be calculated, as was seen previously.
Section 200 edit
Another method is to use the eclipse of the sun or moon in two locations in a straight line, whereby the eclipse occurs in one location before the next. The ratio of the time between eclipses to 24 hours is the proportional to the ratio of the distance between the two points to the circumference of the Earth.
Section 201 edit
Another method is to stand upon a very tall mountain, since the sunset at the mountaintop occurs later than on the plain. Measure how much longer it takes to reach sunset [on the mountain compared to the plain]. Then, use an astrolabe in the following position:
Side AB is parallel to the celestial horizon, and side AE is perpendicular to the celestial horizon. View the sun at point D, measuring how many degrees it is under the horizontal top of the astrolabe, which is angle BAC. Subtract the maximum sine from the secant of angle BAC. Then, set up the proportion: The ratio of the result of the subtraction to the maximum sine is proportional to the straight line height of the mountain to half of the Earth’s diameter. Once the diameter is known, the circumference can be calculated based on a ratio of 7 to 22 [Pi], as was shown previously.
Section 202 edit
The proof to this requires a preface. How is it known that sunset occurs later on a mountain top than on a plain? The reason is that on a mountaintop the line of sight is able to extend to a lower angle, like so:
A man standing on the mountain AB over radius BE sees the sun at point D in the same manner as people at point C over radius CE. Therefore, the sunset on the mountaintop occurs at the same time as it does over the radius CE, which us later that when it occurs over radius BE. In so much as this forms angle BEC, and in triangle BEC the radius BE is also the maximum sine, line AC is the tangent of angle BEC, and line ABE is the secant of this angle, subtracting the maximum sine from the secant AE will yield line AB, which is the height of the mountain. If the height of the mountain and the length of the radius are known [and a ratio can then be determined,] it is possible to know how long after sunset on the plain will sunset on the mountaintop occur.[From Hebrew Commentary: Between the two points lies the secant of angle BEC. It can be determined from the secant, using the table of sines, how many degrees is the angle or arc of BC that is between sunset on the mountaintop and sunset on the plain.] Similarly, if one knows the length of the radius and the [sunset] time in the two locations the height of the mountain can be determined [From Hebrew Commentary: Through knowledge of the times one is able to determine other unknowns, as was seen in section 200. One who knows the measure of arc BC or angle BEG can determine, through use of the tables, how many units are needed to function as a secant for that angle, and after the measure of the maximum sine is determined, which is line BE, or half the diameter of the Earth, it can now be seen that the height of the mountain is the difference between the maximum sine and the secant line.]
Section 203 edit
To measure the diameter of the moon’s orbit by its changing position: This method is based on the changing position of the moon relative to a point in the constellations that we see as we view it from the surface of the Earth. In this figure:
circle ABCD is the sphere of the constellations’ orbit, circle EFG is the sphere of the moon’s orbit, and circle HIJ is the Earth itself. Point K is the center of the Earth, and the moon stands at point F. The line extending from the earth’s center past the moon to the constellations, through the use of which these calculations will be based, is line KFB [the 'true position' of the moon relative to the Earth, not the 'apparent position' to the observer]. The line-of-sight line from an observer past the moon to the constellations is line HFC. The difference between them is angle HFK, as was noted in [section 143, per commentary]. Once you have determined A) angle HKF [via section 105, per Hebrew commentary], the angle of the moon's 'true position'; B) angle KHF, which is the supplementary angle of the angle of 'apparent position' [against the constellation from the viewer's perspective, which is angle AHC]; C) opposite angle HFK, which is the angle of distance [between points B and C is known since the previously mentioned angles are known]; D) and line HK, which is the radius of the Earth [known via section 198, per commentary], then you also know line HF, which is the distance from the observer to the moon, and line KF, which is the distance from the center of the Earth to the moon.
Section 204 edit
After the distance from the center of the Earth to the moon has been determined, calculate how many degrees the moon currently stands in the circumference of its minor orbit. This is determined by the methods in section 150-153, whereby one determines all three angles HFC, HCF, and CHF in the figure in that section. [This figure matches the geocentric model, which views the moon as in larger orbit around the Earth as well as a minor orbit around the line of circumference of its orbit. This allows for calculation of the moon’s actual orbit, which is elliptical, rather than circular. Sometimes the moon is closer to Earth, sometimes it is farther away.] Recall that the length of line HF is already known, as the last section has shown, and HC, the line from the center of the Earth to the moon, can now be known, as can line CF [both of which end at point C on the circumference of the minor orbit]. Line CF is the radius the moon's minor orbit circle.
Section 205 edit
Once the distance from the center of the Earth to the moon's minor orbit is known, the radius of the moon's orbit is known, whose center is removed from the center of the Earth [as stated, the center of the moon's orbit is not the center of the Earth], based on what was written in section 144. Then, after the length of the diameter of the moon's orbit [that is, the ‘virtual circular orbit’ based on the current distance of the moon from the Earth at the moment] is known, the circumference can be calculated, as has been shown previously. Once the circumference is known, one can calculate the diameter of the moon itself by setting up a proportion between the diameter of the moon [in degrees] and the circumference of its orbit, based on the degree measurement of the moon within the circumference of its orbit.
Section 206 edit
To know the degree measure of the moon in its orbit use an astrolabe, as has been previously shown. After this one can calculate in relation to the Earth's center, as was shown in section 203, and then calculate in relation to the center of the moon's orbit, as has been shown.
Section 207 edit
Determining the celestial meridian [IS THIS CORRECT?], upon which this entire calculation was based, requires the use of a magnet. It is a known property of a magnet to point to the north [and the meridian runs from north to south]. Similarly, if you know the location of the geographical pole this can also be determined, since it [the meridian] passes [in a line] directly overhead and through the geographical pole. [IS THIS CORRECT?]. Absent these, one may inscribe a circle on a board and place it flat, parallel to the celestial horizon. A peg is affixed to the center, and the place where the peg's shadow intersects with the circumference at sunrise and sunset is marked off. A line is drawn connecting these two points, and then this line is bisected by a perpendicular line. That line shows the direction of the celestial meridian [running north to south]. For example, in the accompanying figure:
A small peg is placed at point K. At sunrise, its shadow is runs along a line from K to between F and J, and at sunset from K to between points I and B. Line BF can be drawn across, and then line HKD is drawn perpendicular. That line represents the celestial meridian.
Section 208 edit
Another method is to wait until before midday, and using the circle and the peg, observe the last moment before midday that the shadow touches a point on the circumference and note that point. Then, after midday, wait until the first instance of the shadow touching the circumference occurs, and note that point [NOT SURE OF THIS SECTION]. Connect these two points by a line, and the perpendicular line that bisects this line is the direction of the celestial meridian. For example, in the figure in the previous section, one may have seen that before midday the last point on the circumference touched by the shadow is E, and after midday the first point touched is C. With these two point the line CE is drawn and bisected by line HKD, as was done in the previous section, and this is the direction of the celestial meridian. If one desires he can make many concentric cycles around the center, each within the other [to get results closer to the moment of midday itself], observing when the shadow touches them before midday and after.
Section 209 edit
If one desires to know whether it is before, after or exactly midday on any given day he may observe where the shadow falls across the inscribed circle [assuming the board it is inscribed on has not been moved after the previous observation]. If the shadow falls after line HKD it is before midday, if it falls before it is after midday. If it falls on the line itself it is midday. If one wishes to know the same information during the night he should use the same method with the moon provided the moon is in 'Nikud' [HEBREW TERM, COMMENTARY UNSURE OF TRANSLATION, MAY BE SYZYGY, WHEN THE MOON IS IN CONJUNCTION OR OPPOSITION, AS THAT SEEMS TO MATCH THE NEXT CLAUSE], or by the use of a planet or star if they are found in the sky opposite the sun 180 degrees.
Section 210 edit
To know whether the season of ‘Nisan’ (spring) or ‘Tishrei’ (Fall) has arrived one need only to bisect line HKD with a perpendicular line, AKG [text has it as ‘line ABG’, but that does not seem correct], which would be parallel with lines BF and CE. The day that the shadow at sunrise or sunset reaches this line will be the [beginning of the] season of spring or fall.
Section 211 edit
The closer the moon is to the Earth the greater the change of position in respect to the backdrop is when using the calculation method in section 203. This is because the proportion of line HK to line FK is equal to the proportion of the sine of angle HFK to the sine of angle KHF. Since this is so, and reduction in the length of line FK will enlarge angle HFK.
Section 212 edit
Similarly, the closer the moon is to the celestial horizon the greater its apparent change of position will be. This is because the proportion of the lines to each other is equivalent to the proportion of the angle’s sines to each other, as was written in the previous section. As the width of angle HKF decreases its sine will increase, as was noted in section 82. Therefore, the closer it is to the celestial horizon the larger the sine of angle KHF will be, and the larger angle HFK will be. The result is that at the horizon the change in perception will be at its greatest while overhead there will be no perceptible change at all.
Section 213 edit
In order to determine this change at every angle possible from overhead to the horizon one must know the distance of the moon from the center of the Earth, which is line KF; the radius of the earth, line HK; the moon’s distance in degrees from overhead, which is angle HK, or the complementary angle, angle KHF. These values allow one to calculate the remaining necessary values.