6
“To Ease and Expedite the Work”: Mathematical
Instruments and How to Use Them
NO ONE WANTS TO READ EXCERPTS FROM AN OLD CALCULATOR MANUAL. YET instruments have been important throughout the history of mathematics, and a very important and prominent dimension of the mathematics writing of the past is writing about mathematical instruments of one kind or another: making them, using them, where to buy them or have them repaired. It would be a pity to miss out on this material, but of course it presents the obvious problem that quite a lot of it is simply incomprehensible unless you have the instrument in question in your hands.
So this chapter is a compromise. It includes sections on the construction of several classic types of instruments: compasses, sundials, telescopes, “Napier’s bones,” and the “nocturnal” or star clock. It has somewhat less about their use. I hope that in this way you’ll learn something worthwhile about what these instruments were (and are) like, without being faced with material that is impossible to understand.
Today’s mathematical instruments are arguably calculators and computers (and here there’s no chance of giving a passage about how to make one), or, indeed, computer software. As a representative of this kind of modern mathematical “instrument,” I have included a short passage from Peter Duffett-Smith’s Easy PC Astronomy, illustrating the way that these instruments allow us to perform with ease calculations which by more traditional means would be slow, complex, and inaccurate to the point of infeasibility.
“Cards for the Sea”
Martín Cortés (1532–1589) was the son of Hernán Cortés, the first of the conquistadors. He published his account of navigation (Breve compendio de la sphera y de la arte de navegar . . .) in Spanish in Seville in 1551, and a decade later it received this English translation by Richard Eden. Eden (c. 1520–1576) was the translator of various works on subjects including science and exploration, and a promoter of colonial ideas.
Various instruments and mathematical techniques were described in Cortés’ book; here we see a description of one of the most basic: how to draw a compass rose, the starting point for both map making and accurate direction finding at sea.
Martín Cortés, trans. Richard Eden, The Arte of Nauigation, Conteynyng a compendious description of the Sphere, with the makyng of certen Instrumentes and Rules for Nauigations: and exemplified by many Demonstrations. Wrytten in the Spanyshe tongue by Martin Curtes, And directed to the Emperour Charles the fyfte. Translated out of Spanyshe into Englyshe by Richard Eden. (n.p., 1561), lvir–lviir.
Arriving to the end desired (which is navigation, the principal intent why I began this work), I say that Navigation or sailing is none other thing than to journey or voyage by water from one place to another, and is one of the four difficultest things whereof the most wise king hath written. 
These voyages being so difficult, it shall be hard to make the same be understood by words or writing. The best explication or invention that the wits of men have found for the manifesting of this is to give the same painted in a Card, for the draught or making whereof, it shall be requisite to know two things: whereof the one is the right position of places, or placing of countries and coasts; the other is the distances that 
from one place to another. And so the Card shall have two descriptions. The one that answereth to the position shall be of the winds, which the Mariners call lines or points of the compass; and the other, that answereth to the distances, shall be the drawing and pointing of the coasts of the land and of the Islands compassed with the sea.
To paint the winds or lines, you must take skins of parchment or large paper, of such bigness as you will the Card to be. And in it draw two 
lines with black ink, which in the midst shall cut or divide themselves in right angles, the one according to the length of the Card, which shall be East and West, and the other North and South. Upon the point where they cut, make a center, and upon it, give a privy or hid circle which may occupy in manner the whole Card. This circle some make with lead 
, that it may be easily put out. These two lines divide the circle into four equal parts. And every part of these shall you divide in the midst with a prick or punct. Then from one punct to another, draw a right Diametrical line with black ink: and so shall the circle remain divided with four lines into eight equal parts, which correspond to the eight winds. In like manner shall you divide every 
of the eight into two equal parts: and every part of these is called a half wind. Then draw from every punct to his opposite diametrically, a right line of green or azure. Likewise shall you divide every half wind in the circle, into two equal parts. And from these puncts which divide the quarters, you shall draw certain right lines with red ink, which also shall pass by the center, which they call the mother compass or chief compass of the Card, being in the midst thereof. And so shall come forth from the center to the circumference 32 lines, which signify the 32 winds.
Beside these said lines, you shall make other
equal distant to them, and of the self same colours, in this manner. From the points of the winds and half winds that pass by the center, draw certain right lines that pass not by the center, but be equally divided to those that pass by the center, and of the same colours and equidistance as are they that pass by the center. And as these lines concur together as well in the center as in the points of the winds and half winds that are in the circumference of the circle, they shall leave or make there 
other 16 compasses, every one with his 32 winds. And if the Card be very great, because the lines may not go far in sunder, if you will make there other 16 compasses, you must make them between the one and the other of the first 16 points, where the quarters are made with their winds, as we have said.
It is the custom for the most part to paint upon the center of these compasses a flower or a rose, with diverse colours and gold, differencing the lines, and marking them with letters and other marks: especially signing the north with a fleur de lys, and the East with a cross. This, beside the distinction of the winds, serveth also for the garnishing of the Card. And this for the most part is done after that the coast is drawn. And thus much sufficeth for the draught of the winds.
Making a Horizontal Sundial
The making of sundials has been the subject of many, many treatises, and represents one of the oldest and most widespread practical uses of geometry. Our extracts are from one of the earliest such books to be printed in English, by Thomas Fale, a clergyman. It was Fale’s only book and was reprinted in 1626 and 1652.
Thomas Fale (1561–after 1604), Horlogiographia The art of dialling: teaching an easie and perfect way to make all kinds of dials upon any plaine plat howsoeuer placed: vvith the drawing of the twelue signes, and houres vnequall in them all. Whereunto is annexed the making and vse of other dials and instruments, whereby the houre of the day and night is knowne. Of speciall vse and delight not onely for students of the arts mathematicall, but also for diuers artificers, architects, surueyours of buildings, free-Masons and others. (London, 1593), 4v–5v.
The making of a Horizontal or plain lying Sundial.
Your plate being prepared smooth and plain, draw upon it two lines as in the figure following (6.1), the one AB, the other CD, cutting themselves squarewise: that is, making right angles in the point E. Upon which, make the quadrant of any circle from the line EC to the line EA or EB, and write at C the North, at D the South, at A the East, at B the West. And the line CA, which here is the quadrant, being divided into 90 degrees or parts, the elevation of the Pole shall be accounted in it (which in our example is 52 degrees) from C to A; and at the end of this number draw a line from the centre E, which shall be EF, representing the style and artery of the world.
Then draw another line KL by C, or by some other point of the line DC, squarewise, so long as you can, which shall be called the touch line, or line of Contingence. Then measur
with your compasses the least distance of the point O and the line EF or the Style. 
the one foot placed in O, which is the point of intersection, and the other extended toward E, where it shall chance to divide or be placed in the line EC, mark that point or centre with the letter G. And draw with your compasses a half circle upon this centre for the equinoctial circle, from H by C to J, whose diameter must be equally distant to the line LK. Then divide this half circle into twelve equal parts.
Figure 6.1. The design for a plain lying sundial. (Fale, fol. 5v. © The Bodleian Libraries, University of Oxford. 4 F 2 Art.BS., p. 5.)
This done, lay a ruler upon the centre G, and upon every mark or division made in the half Equator, and where the ruler shall touch the line of contingence, there make marks or pricks, by which pricks draw lines from E for the hours. EC is the twelfth hour, EB the sixth in the morning, EA the sixth at evening; the rest you may see in the figure.
And whereas in Summer the fourth and fifth in the morning, and also the seventh and eighth at evening, shall be necessary in this kind of Dial, prolong or draw the lines of four and five at evening, beyond the centre E, which shall show the hours of four and five in the morning. And likewise the seven and eight in the morning, for the seven and eight at evening.
You may observe an order both in these and in all other erect direct dials, by dividing the one half of the Equator, drawing hour lines for the forenoon, and observing the same distance from the Meridian line, on the other side for the afternoon; for the line of the eleventh hour in the forenoon is of like distance from the Meridian, that the first is in the afternoon, and the tenth as 
second, and so of the rest.
When you would draw or make the half hours, you must divide every part of the Equator into two equal parts, using the ruler and the line of contingence as you find in the drawing of the hour lines.
And this remember for the drawing of the half-hour lines, not only in this kind, but also in all other kinds of dials, which afterward shall follow: the Style must be fixed in the centre E, hanging directly over the Meridian line EC with so great an angle as the lines CEF make, declining from that on neither side.
The Equinoctial circle, the Quadrant, 
the line of the Style and of Contingence must be lightly drawn, because they ought to be put out again, in that they serve to no use but for the drawing of the Dial. And this likewise remember in all other kinds of Dials: that the preparative or pricked lines must, after the making of the Dial, be omitted and extinguished, as altogether unprofitable.
This and all other kinds of Dials may most fitly be drawn upon a clean paper, and then with the help of your compasses placed on the plate.
Speaking-Rods
The Scottish mathematician John Napier (1550–1617), better known today as the inventor of logarithms, also devised a system of rapid computation known as “Napier’s bones.” He first published a description of them in Latin in 1617, a translation followed in 1627, and many later publications elaborated on them. Seth Partridge, the author of this extract, was a mathematics teacher with a particular interest in mathematical instruments; his description of the “speaking rods” was originally written for his students.
Seth Partridge (1603/4–1686), Rabdologia, or, The Art of numbring by Rods, whereby the tedious operations of Multiplication, and Division, and of Extraction of Roots, both Square and Cubic, are avoided, being for the most part performed by Addition and Subtraction: with Many Examples for the practise of the same: First invented by the Lord Napier, Baron of Marchiston, and since explained, and made usefull for all sorts of men. By Seth Partridge, Surveyor, and Practitioner in the Mathematics. (London, 1648), pp. 1–7, 22–23.
Speaking-Rods, their Definition and Fabric.
RABDOLOGIA is the Art of counting by Numbering-Rods. As it pleased that ever famous Author the Lord Napier, the first Inventor of that most admirable invention of Logarithms, to call those his excellent tables by the name of Logarithms, that is to say, speaking-Numbers, even so this of the Rods, Rabdologia, that is to say, speaking-Rods, or the Speech of Rods. And speaking-Rods they are very properly called, for being so Tabulated, that is to say, placed, set, and laid together, as their nature requireth, they do of themselves tell us, or show forth unto us, with unexpected ease and certainty, and without any operation at all, what the product in any Multiplication is, and what the quotient in any division is, without any charge or trouble of memory, and in Extraction of the Square and Cube root, they do very much ease and expedite the work.
These speaking-Rods may be made either of Silver, Brass, Ivory, or Wood, as the maker and user of them best pleaseth, but they are most ordinarily made of good solid Box
, and being thereof made, they are as useful as those made of any other substance whatsoever. Nay, I hold them more light and nimble then those made of Metal.
They are foursquare pieces, all placed to a just and even thickness, near about 
of an inch square, or something better, and their length is just nine times their breadth, so that they be two inches, or near thereupon, in length. The number of these foursquare Rods are most ordinarily Ten, which are enough for ordinary use; you may have 20, 30, or 40 if you will, and as your occasion serveth. To these Rods, how many soever you have, belongeth a broad piece, of the same substance that the Rods be of, called Lamina, and is of the same length and thickness with the Rods, and the breadth is near about the breadth of three of the Rods.
These Rods, being made thus foursquare, have each of them four faces, so that 10 Rods have on them 40 faces, which four faces of every Rod are divided into Nine equal parts, by Nine lines drawn round about the Rod, so that upon each face are Nine perfect squares made. And 
of those 9 squares or parts are divided into two parts by Diagonal lines. And in those Divisions, or squares, are inscribed Arithmetical figures, beginning at the top, or first square division, and from thence descending downwards to the Ninth Division in an Arithmetical progression, so that they are by this means fitted ready for many Arithmetical works, as Multiplication, Division, and the extraction of Roots, etc.
Figure 6.2. The faces of the speaking-rods. (Napier, facing p. 7. © The Bodleian Libraries, University of Oxford. 8° R 84 Med.)
But to avoid many words in describing these vocables, behold the shape and figure of all those 10 Rods, and the numbers set on every of their four faces (Figure 6.2). Every practitioner may make them himself by cutting the faces of every one of the printed papers , and so plac
on a square piece of wood as before. Or else they are ready made in Wood, by Master John Thompson in Hosier Lane near Smithfield, who makes all kind of Mathematical Instruments, and also by Mr Anthony Thompson in Gresham College, and by Mr Thomas Browne at the Globe near Aldgate. In Silver or Brass they are made by Mr Elias Allen, over against St Clements without Temple Bar.
But note, by the way, that though I have inscribed upon the Rods the same numbers that were used at the first invention, yet I have not inscribed them altogether in the same manner. For there the inscriptions upon the third and fourth faces were set on inverse unto those numbers on the first and second faces. And this way of inscription I find to be the best, because it saveth the labour of turning the Rods end for end, which of necessity must be done when the numbers are set on them inverse, according to that other way.
In this description of the Rods, you may see the figures of each face of the Rod do begin at the top, or upper square space next to the end, and descend down in an Arithmetical progression. As, on the second face of the first Rod, the figures begin at the top with an unit—thus: 1—on the right of the Diagonal line, and descend downward in an Arithmetical progression—thus: 1, 2, 3, 4, 5, 6, 7, 8, 9—every figure downwards increasing by one unit. On the third face of the same Rod the upper figure is 9, and the rest of the figures, descending downwards, being 18, 27, 36, 45, 54, 63, 72, 81, each square exceeding his next by 9. And on the fourth face the figures are 8, 16, 24, 32, 40, 48, 56, 64, 72, every number equally exceeding or increasing his former by 8. But upon the first face of the same first Rod are only ciphers, as there are also upon one of the faces of the second, third, and fourth Rods.
There is also to these Rods a frame to be made, to Tabulate or lay the Rods in when you work with them (you may use them without if you please, but to work with them in a frame is far the better way, and is always intended to be used with them through all this following discourse). This frame is but only a small thin board, as large for length as is the length of the Rods, and as wide as the breadth of all the 10 Rods, with the Lamina, laid all close together side by side; but if you have a greater number of Rods than 10, it is requisite the frame should be the larger, that so you may Tabulate more than 10 places of figures in one number. This board is to have two ledges upon it of equal thickness to the Rods set on squarewise, the one at the top, and the other at the left side, and that ledge on the left side is to be divided into nine equal square parts as the Rods are, and figures set on them, beginning at the head or top square, and descending downwards, thus: 1, 2, 3, 4, 5, 6, 7, 8, 9, like to the second face of the first Rod. It is far better to Tabulate the Rods in a frame, or Table thus made, than without, for the frame keeps them even at the head, and to lie square and close together, and the numbers on the ledge do very readily guide you to the product, in any Multiplication, and to the figure for the quotient, in any Division.
Of Multiplication by the Rods.
Multiplication is either single or compound.
Single Multiplication is when the Multiplicator consisteth but of one single figure only, though the multiplicand consist of many.
Compound Multiplication is when the numbers to be multiplied together, that is to say, the Multiplicand and the Multiplicator, consist, both of them, of more places of figures than one.
For single Multiplication by the Rods, you are first to Tabulate the Multiplicand 
, and then look upon the left hand ledge of the frame for the figure of your Multiplicator, against which figure, you have in the line of squares upon the Rods the product sought for.
Example: if you are to multiply 672 by 4. Tabulate three Rods that have on their top squares the three figures 6, 7 and 2, the three figures of the number given, in order as they ought. First, place the Rod with the figure 6 next the ledge, and the 7 next, and the Rod with the figure 2 last. Which being done, look upon the ledge for the figure 4, against which, upon the fourth line of squares on the Rods, you shall see in the first half-Rhomboids 2, and in the first whole Rhomboids 4 and 2, that is, 6. In the second Rhomboids, 8, and, lastly, in the last and outermost half-Rhomboids, 8. So that you must set down the sum of that rank, or line of numbers, thus: 2688, which is the true product of 672 multiplied by 4, which was demanded.
Telescopes Refracting and Reflecting
The Juvenile Library appeared in monthly installments during 1800–1801, and thereafter under the title of The Juvenile Encyclopedia until 1803; the collected edition eventually filled six volumes. It promised instruction in a wide range of “useful” subjects, and if that were not enough, its anonymous authors also publicized it through a mildly hysterical prospectus (“. . . the youthful genius of the whole nation . . .”) and offered monthly prizes (books, scientific instruments, or both) and the chance for the best “student productions” to appear in print. The telescopes described here were among the prizes.
Figure 6.3. The refracting telescope. (The Juvenile Encyclopedia, vol. 3, 3rd plate of “Optics.” © The Bodleian Libraries, University of Oxford. (Vet.) 3985 e.19–24.)
The Juvenile Encyclopedia. Including a Complete Course of Instruction, on every useful subject: Particularly Natural and Experimental Philosophy, Moral Philosophy, Natural History, Botany, Ancient and Modern History, Biography, Geography and the Manners and Customs of Nations, Ancient and Modern Languages, English Laws, Penmanship, Mathematics, and the Belles Lettres, vol. 3. (London, 1801), pp. 130–132.
What microscopes perform upon minute bodies very near, telescopes perform upon great bodies very remote; namely, they enlarge the angle in the eye under which the bodies are seen, and thus, by making them very large, they make them appear very near; the only difference is that in the microscope the focus of the glasses is adapted to the inspection of bodies very near, in the telescope to such as are very remote. Suppose a distant object at AB (see Figure 6.3), its rays come nearly parallel, and fall upon the convex glass cd; through this they will converge in points, and form the object E at their focus.
But it is usually so contrived, that this focus is also the focus of the other convex glass of the tube. The rays of each pencil,° therefore, will now diverge before they strike this glass, and will go through it parallel, but the pencils all together will cross in its focus on the other side, as at e, and, the pupil of the eye being in this focus, the image will be viewed through the glass, under the angle geb, so that the object will seem at E under the angle DeC.
This telescope inverts the image, and therefore is only proper for viewing such bodies as it is immaterial in what position they appear, as the fixed stars, etc. By adding two convex glasses, the image may be seen upright. The magnifying power of this telescope is found by dividing the focal distance° of the object-glass by the focal distance of the eye-glass, and the quotient expresses the magnifying power.
Figure 6.4. The reflecting telescope. (The Juvenile Encyclopedia, vol. 3, 3rd plate of “Optics.” © The Bodleian Libraries, University of Oxford. (Vet.) 3985 e.19–24.)
An inconvenience was found to attend the use of this instrument, as when any extraordinary magnifying power was wanted the field of view, and even the image, was found to be tinged with different colours. The reason of this will be plain, when I come to treat of the prism and the prismatic colours. You will then see that if a lens is very convex, the edge acts like a prism, and separates the component particles of light, which are differently coloured, and consequently a round circle of different coloured rays is produced. To remedy this, Mr. Dolland,° finding that flint and crown glass had different refracting powers, and that crown glass (the common window glass) dispersed the rays of light less than any other, adapted two convex glasses of crown glass to a double concave of flint glass (which has the greatest dispersive power) so as exactly to fit, and by that means made them counteract each other, so that the field of view is presented perfectly colourless. These telescopes, therefore, are called achromatic (or colourless) telescopes.
The reflecting telescope performs, by reflecting the rays issuing from any object, what the last did by refracting them. Let ab (see Figure 6.4) be a distant object to be viewed; parallel rays issuing from it, as ac and bd, will be reflected by the metallic concave mirror cd to st, and there brought to a focus, with the image a little further and inverted, agreeably to the effect of a concave mirror on light . The hole in the mirror cd does not distort or hurt the image st, it only loses a little light. Nor do the rays stop at the image st; they go on, and cross, a little before they reach the small concave mirror en. From this mirror the rays are reflected nearly parallel through the hole O, in the large mirror, to R; there they are met by the plano-convex lens hi, which brings them to a convergence at S, and paints the image in the small tube of the telescope close to the eye. Having, by this lens, and the two mirrors, brought the image of the object so near, it only remains to magnify this image by the eye-glass kr, by which it will appear as large as zy.
To produce this effect, it is necessary that the large mirror be ground so as to have its focus a little short of the small mirror, as at q, and that the small mirror should be of such concavity as to send the rays a little converging through the hole o, that the lens bi should be of such convexity as to bring those converging rays to an image at S, and that the eye-glass kr should be of such a focal length, and so placed in the tube, that its focus may just enter the eye through the small hole in the end of the tube.
To adapt the instrument to near or remote objects, or rather to rays that issue from objects converging, diverging, or parallel, a screw, at the end of a long wire, turns on the outside of the tube, to bring the small mirror nearer to, or farther from, the large mirror, and so as to adjust their focuses according to the nearness or remoteness of the objects.
To estimate the magnifying power of the reflecting telescope, multiply the focal distance of the large mirror by the distance of the small mirror from the image S; then multiply the focal distance of the small mirror by the focal distance of the eye-glass kr; then divide these two products by one another, and the quotient is the magnifying power.
Notes
pencil: a set of lines passing through a single point; so, in this case, the set of light rays emerging from a single point.
focal distance: the distance from the centre of the lens to its focus.
Mr. Dolland: John Dolland (1706–1761), a British optician who was awarded the Royal Society’s Copley Medal for his work on achromatic lenses, although he was not the first to make them.
Scales Simple and Diagonal
Originally published in 1849, this treatise promised “to put within the reach of all” a description of the scientific instruments of the day. Many subsequent editions followed, and the book was used in British military and naval schools; the author taught at the Royal Military Academy at Woolwich. By the time of this, the fourteenth edition it even, according to the preface, “formed, by authority, part of a midshipman’s kit.”
J. F. Heather (d. 1886), A Treatise on Mathematical Instruments: Their construction, Adjustment, Testing, and Use Concisely Explained. Revised, with Additions By Arthur T. Walmisley, M.I.C.E. (London: 14th edition, 1888), pp. 9–11, 84–86.
Scales of equal parts are used for measuring straight lines, and laying down distances, each part answering for one foot, one yard, one chain, etc., as may be convenient, and the plan will be larger or smaller as the scale contains a smaller or a greater number of parts in an inch.
Scales of equal parts may be divided into three kinds: simply divided scales, diagonal scales, and vernier scales.
Simply-divided Scales.
Simply-divided scales consist of any extent of equal divisions, which are numbered 1, 2, 3, etc., beginning from the second division on the left hand. The first of these primary divisions is subdivided into ten equal parts, and from these last divisions the scale is named. Thus it is called a scale of 30, when 30 of these small parts are equal to one inch. If, then, these subdivisions be taken as units, each to represent one mile, for instance, or one chain, or one foot, etc., the primary divisions will be so many tens of miles, or of chains, or of feet, etc.; if the subdivisions are taken as tens, the primary divisions will be hundreds; and, if the primary divisions be units, the subdivisions will be tenths.
The accompanying drawing (Figure 6.5) represents six of the simply-divided scales, which are generally placed upon the plain scale. To adapt them to feet and inches, the first primary division is divided duodecimally upon an upper line. To lay down 360, or 36, or 3.6, etc., from any one of these scales, extend the compasses from the primary division numbered 3 to the sixth lower subdivision, reckoning backwards, or towards the left hand. To take off any number of feet and inches, 6 feet 7 inches for instance, extend the compasses from the primary division numbered 6, to the seventh upper subdivision, reckoning backwards, as before.
Figure 6.5. Simply-divided scales. (Heather, p. 9. © The Bodleian Libraries, University of Oxford. 1876 f.2.)
Figure 6.6. Diagonal scales. (Heather, p. 10. © The Bodleian Libraries, University of Oxford. 1876 f.2.)
Diagonal Scales.
In the simply-divided scales one of the primary divisions is subdivided only into ten equal parts, and the parts of any distance which are less than tenths of a primary division cannot be accurately taken off from them; but, by means of a diagonal scale, the parts of any distance which are the hundredths of the primary divisions are correctly indicated, as will easily be understood from its construction, which we proceed to describe.
Draw eleven parallel equidistant lines (see Figure 6.6); divide the upper of these lines into equal parts of the intended length of the primary divisions; and through each of these divisions draw perpendicular lines, cutting all the eleven parallels, and number these primary divisions, 1, 2, 3, etc., beginning from the second.
Subdivide the first of these primary divisions into ten equal parts, both upon the highest and lowest of the eleven parallel lines, and let these subdivisions be reckoned in the opposite direction to the primary divisions, as in the simply-divided scales.
Draw the diagonal lines from the tenth subdivision below to the ninth above, from the ninth below to the eighth above, and so on, till we come to a line from the first below to the zero point above. Then, since these diagonal lines are all parallel, and consequently everywhere equidistant, the distance between any two of them in succession, measured upon any of the eleven parallel lines which they intersect, is the same as this distance measured upon the highest or lowest of these lines, that is, as one of the subdivisions before mentioned. But the distance between the perpendicular, which passes through the zero point, and the diagonal through the same point, being nothing on the highest line, and equal to one of the subdivisions on the lowest line, is equal (Euclid 
6, proposition 4)° to one-tenth of a subdivision on the second line, to two-tenths of a subdivision on the third, and so on; so that this, and consequently each of the other diagonal lines, as it reaches each successive parallel, separates further from the perpendicular through the zero point by one-tenth of the extent of a subdivsion, or one-hundredth of the extent of a primary division. Our figure (6.6) represents the two diagonal scales which are usually placed upon the plain scale of six inches in length. In one, the distances between the primary divisions are each half an inch, and in the other a quarter of an inch. The parallel next to the figures numbering these divisions must be considered the highest or first parallel in each of these scales to accord with the above description.
The primary divisions being taken for units, to set off the number 5.74 by the diagonal scale. Set one foot of the compasses on the point where the fifth parallel cuts the eighth diagonal lines, and extend the other foot to the point where the same parallel cuts the sixth vertical lines.
The primary divisions being reckoned as tens, to take off the number 46.7. Extend the compasses from the point where the eighth parallel cuts the seventh diagonal to the point where it cuts the fifth vertical.
The primary divisions being hundreds, to take off the number 253. Extend the compasses from the point where the fourth parallel cuts the sixth diagonal to the point where it cuts the third vertical.
Now, since the first of the parallels, of the diagonals, and of the verticals indicate the zero points for the third, second, and first figures respectively, the second of each of them stands for, and is marked, 1, the third, 2, and so on, and we have the following.
General Rule.
To take off any number to three places of figures upon a diagonal scale. On the parallel indicated by the third figure, measure from the diagonal indicated by the second figure to the vertical indicated by the first.
Note
Euclid book 6, proposition 4, states that if two triangles have the same set of angles, then their sides are proportional, with corresponding sides being opposite corresponding angles.
Making a Star Clock
Roy Worvill published a number of guides to amateur astronomy in the 1960s and 1970s, including some for young children. Here he gives a wonderfully homely account, complete with cardboard and paper fastener, of the instrument sometimes called a “nocturnal” but here evocatively dubbed the “star clock.” It has its roots in the books on “dialing” of the sixteenth and seventeenth centuries, such as Thomas Fale’s, above.
Roy Worvill (1914–2003), Telescope Making for Beginners (London, 1974, 1976), pp. 70–73.
Making a Star Clock
The sun-dial has the obvious drawback that it is quite useless when the sky is cloudy and at any time when the sun is below the horizon. Indeed, before clocks were invented a cloudy sky made time-keeping a difficult operation since clouds hide the stars as well as the sun. Nevertheless, there are many clear nights, even in the uncertain climate of Britain, and the star clock or nocturnal, to use its old name, then came into its own.
The east-to-west movement of the stars, like that of the sun, is an appearance produced by the earth’s daily rotation, and so the stars can be used at night for telling time as the sun can be by day.
In some respects the stars have an advantage, for there are star-groups, or constellations, which never sink below our horizon. To find these we must look towards the northern part of the sky. There, at a height above our horizon equal to our latitude, measured in degrees, we shall find the Pole Star. In fact the Pole Star is not exactly at the north pole in the sky, but it is very near it. The Pole Star appears to remain still, with the other stars circling about it every twenty-four hours as the earth carries us round. Since the Pole Star does not mark the position of the pole exactly it does, in fact, make a very small circle itself. But for our present purpose we can regard it as fixed.
Around the Pole Star there are a number of constellations which never set. These are called the circumpolar groups. The most familiar of them are the seven bright ones which make up the Plough or Dipper, though they are actually part of a much larger constellation called the Great Bear, or Ursa Major, to use its Latin name. The seven stars of the Plough are the hand of our clock and the sky itself is the dial. In particular we use the two stars called the Pointers which lie at the outer edge of the group, the side of the Dipper or saucepan farthest away from the handle. These stars get their name because they point roughly to the Pole Star.
We shall find them rather low down in the northern sky if we look for them in the evening during autumn. In spring, on the other hand, they are to be found high overhead, even if we look for them at the same hour. The reason for this is that the earth travels round the sun a certain distance each day. Measured in miles it is a very long way, for we are moving at just over 18 miles per second, but measured in degrees it is only about one degree daily. The complete circuit of the sun, which takes a year, represents a turn of 360 degrees, and of this a tiny fraction under one degree is covered every day. As a result of these two movements the stars of the Plough make one complete turn round the Pole Star every day as the earth spins on its axis, and a little bit more, approximately one degree, as a result of the other turning movement caused by the earth’s revolution round the sun. Because of the little extra movement we find that from night to night, if we look for them at exactly the same time, they have moved a little farther in their circular track round the celestial pole. This complicates our star clock a little, but not too much, since we can make the necessary adjustment for the date. It is, however, rather like having a clock which not only moves its hands but also has a dial which moves slowly round once in a year.
To make the star clock we need the following: a piece of cardboard about sixteen inches long and eight inches wide, a round-headed paper fastener (not a wire clip), a pair of compasses, scissors, a knife and a cutting-board. If you are handy with a fret-saw you can make a better one by using plywood or some other material more rigid than cardboard. Some of the ones made centuries ago were of brass and beautifully engraved and decorated. These are still to be found in museums with collections of old scientific instruments.
Figure 6.7. The construction of the star clock. (Worvill, p. 72.)
To make the nocturnal, first cut the sheet of cardboard into two equal squares, each having sides of eight inches. From one of the two squares a circle of about three inches radius is cut out and a small “window” hole, about half an inch square, is cut in the position shown in the diagram (Figure 6.7) between two of the radius lines which mark out angles of sixty degrees. In the sector of the circle which is diametrically opposite the window hole draw in the straight line joining the two ends of the diameters. This line will always indicate the position of your horizon. The centre of the circle shows the position of the celestial pole, marked approximately, as we have seen, by the Pole Star. The seven stars of the Dipper are drawn in each of the six sectors of the circle as shown in Figure 6.7, indicating the movement they follow in the course of every twenty-four hours.
Figure 6.8. The construction of the star clock, continued. (Worvill, p. 73.)
On the other cardboard square draw the five concentric circles shown in Figure 6.8, the largest one of radius four inches and the others, proceeding inwards, of three and a half inches, three inches, two and a half inches and two inches. Draw a diameter to the outer circle and mark off angles of thirty degrees so that there are twelve spaces or sectors. On the outer circle these are marked with two-hourly divisions from twelve o’clock midnight throughout the twenty-four hours, although of course the nocturnal can only be used when it is dark enough to see the stars.
Place the small cardboard disc centrally over the large one and make a mark on the lower one through the small window hole. In the ring where the mark appears write the months of the year, the dates showing the seventh day of the month. March 7th should be marked on the line which shows midnight on the hour circle. That is the time when the two stars of the Pointers are in a vertical line with your horizon. You will see from the diagram that the months are marked from March onwards following a clockwise direction while the hours from midnight go anti-clockwise.
The smaller circle is then attached with a paper fastener, through a small central hole to the larger one so that it will revolve easily, but not too loosely, over the lower dial. This completes the making of the star clock.
To use it, you hold it in front of you so that the centre of the dial, marked by the fastener, is approximately over the Pole Star. The smaller dial is turned so that it shows the date, as nearly as possible through the small window hole. The whole clock is then turned so that the horizon line is level at the lowest position of the circle. Look at the Dipper stars to see which of the six star diagrams corresponds most accurately to the position of the Dipper. A line from the Pointers in that section of the star diagram is followed to the outside edge of the dial, and this will show you the time. By this means it should be possible to tell the time to within about ten or fifteen minutes. Of course your watch will do this more accurately, but even the best watches can run down or go wrong, and it is interesting to remember that the first clocks and watches were so unreliable that a small pocket sun-dial or nocturnal was often carried in those days to check the time-keeping of the watch or clock! Apart from any practical use as a time-keeper the nocturnal, like the sun-dial, will give you a better understanding of the changing appearance of the sky through the seasons.
PC Astronomy
A follow-up to earlier books by the same author—Astronomy with Your Pocket Calculator and Astronomy with your Personal Computer—this volume, Easy PC Astronomy, was geared toward the efficient and convenient making of sometimes complex astronomical calculations rather than toward producing glossy displays or pictures. It discussed a wide variety of often quite technical computations connected with the practice of amateur astronomy and continued the practical theme we have seen in the making of sundials and star clocks.
Peter Duffett-Smith (dates unknown), Easy PC Astronomy (Cambridge, 1997), pp. 48–51. © Cambridge University Press 1997; reproduced with permission.
Precession
Anyone who has played with a spinning top as a child will know that its axis slowly gyrates about the vertical direction as it spins. This phenomenon is common to any spinning object which is under the influence of an external couple, and is called gyroscopic precession. In the case of the spinning top, the couple is provided by the weight of the top which pulls vertically downwards through its centre of gravity. The Earth is also a spinning object and it, too, exhibits the same precessional behaviour. The gravitational attractions of the Moon and the Sun, acting on the equatorial bulges of the slightly non-spherical Earth, cause the couple which results in the north–south axis precessing slowly about a line through the centre of the Earth and perpendicular to the plane of the ecliptic° with a period of about 26,000 years. The effect of this luni-solar precession is to make the direction of the line of intersection of the rising node of the celestial equator on the plane of the ecliptic (the first point of Aries of vernal equinox)° move steadily at a rate of about 50 arcseconds per year.
There is also another, smaller, precession from the gravitational influences of the planets. If the celestial equator were fixed (i.e. not perturbed by luni-solar precession), their combined effect would be to cause the equinox to move by about 12 arcseconds per century, and to decrease the obliquity of the ecliptic° by about 47 arcseconds per century. This is called planetary precession. Both luni-solar and planetary precession are usually calculated together as general precession.
The positions of “fixed” stars and other celestial objects are often defined by their right ascensions and declinations, or by their ecliptic longitudes and latitudes. In either case, the reference direction is that of the first point of Aries, so general precession causes these coordinates to change slowly with time. Hence it is necessary to specify the moment, or epoch, at which a particular pair of coordinates is valid. A correction can then be applied to convert the coordinates to values which are valid at another epoch. AstroScript incorporates an algorithm for performing this correction for general precession rigorously it converts the currently held right ascension and declination into their new values. It asks you to supply the two precessional epochs if you have not already done so.
The AstroScript algorithm for general precession follows that laid out in the Explanatory Supplement to the Astronomical Almanac . If (α1, δ1) are the right ascension and declination valid at epoch 
1, the column vector r1 is first obtained by
This is then multiplied by the precession matrix, P, to form the new column vector, r2, appropriate for the new epoch 
2 as follows:
r2 = P · r1.
If the components of r2 are
the new coordinates, (α2, δ2), are obtained from r2 by the equations
The precession matrix P is given by
where cx = cos ζ A, sx = sin ζ A, cz = cos zA, sz = sin zA, ct = cos θA, and st = sin θA. These arguments are calculated from the following expressions involving the interval in Julian centuries of 36525 days between 0, the fundamental epoch J2000.0 (1.5 Jan. 2000), and the two epochs 1 and 2, where
JD() represents the Julian day number of epoch , and JD(0) = 2 451 545.0,
ζA, zA, and θA are all given in arcseconds by the above expressions.
Note
ecliptic: the plane of the earth’s orbit around the sun.
the line of intersection . . .: Another way to describe this is as the line in which the plane of the earth’s orbit and the plane of the equator intersect.
the obliquity of the ecliptic: the angle between the plane of the earth’s equator and that of its orbit.