More on sectors

This paper was originally published in the Slide Rule Gazette, Issue 16, Autumn 2015 by the UK Slide Rule Circle.

In Issue 12 (Autumn 2011) of the Slide Rule Gazettei Peter Hopp discussed a collection of 28 sectors that appeared on eBay featuring eighteen “makers” names. He tentatively concluded from their similar appearance that there were perhaps only one or two actual makers. The sectors examined dated from the late 18th century to the end of the 19th.

I would like to re-examine Peter’s tentative conclusion based on my own collection of 22 British sectors. However before I do so I need to say a little about the evolution of the sector and the differences in the scales between the arch joint sector and the round joint sector.

In the early and mid eighteenth century English sectors were generally made of brass and frequently featured an additional hinged plate (the friction leaf) that functioned as a steady and this is a convenient starting point for my chronology because the scales were already becoming standardised. An illustration of both sides of a sector of this period can be found in the modern reprint of Edmund Stone’s English edition of Bionii. This illustration is reproduced in Figure 1. There is also a description of the construction and use of the English sector.

Figure 1 - 1758 Edmund Stone illustration

The scales on this sector, which I have divided into two types namely sectoral (radiating from a single point) and parallel (to an edge) are as follows:

Upper side
Sectoral: L, the line of equal parts

S, the line of natural sines

ƒ, the line of natural secants

P, the line of polygons

Parallel: S, the line of artificial (logarithmic) sines

T, the line of artificial tangents

Note these last two continue across both halves of the sector when fully opened

Lower side
Sectoral: C, the line of chords

T, the line of natural tangents

t, a continuation of the line of natural tangents

Parallel: In, a foot divided into twelve inches

N, Gunter’s line of numbers (logarithms)

Edmund Stone also observes that there are sometimes extra lines, not shown in the figure, placed in vacant spaces, namely:

Line of hours

Line of latitudes

Line of inclinations of the meridian

Recently I have acquired a brass sector by Thomas Wright, made between 1718 and 1747 that is shown in Figure 2.

Figure 2 - 6” brass sector signed T Wright fecit

The scales on this sector are:
Upper Side
Sectoral: Cho, the line of chords

Lin, the line of equal parts

Sec, the line of natural secants

Poly, the line of polygons

Parallel: N, Gunter’s line of numbers (logarithms)

Lower Side
Sectoral: Sin, the line of natural sines

Tan, the line of natural tangents

tan, a continuation of the line of natural tangents

Parallel: T, the line of artificial tangents

S, the line of artificial sines

Outer Edge: A 12 inch ruler divided to tenths

This is exactly the same set of scales as illustrated in Stone’s work although they are arranged a little differently. In fact the series of scales on English sectors had become standardised by the beginning of the 18th century as noted by Wynter & Turner.iii

About the middle of the eighteenth century a transition gradually took place, ivory replacing brass. These ivory instruments had arch joints in either silver or, more commonly, brass. In 1775 the third edition of a book by John Robertson was published that again includes an illustration of a sector and again there is conveniently a modern reprint.iv

Figure 3 - Illustration in the book by John Robertson

Figure 3 shows the Robertson illustration. There are more scales than shown in Figure 1. These are:

Upper side
Sectoral: Lin, the line of lines or equal parts

Cho, the line of chords

ƒec, the line of natural secants

Poly, the line of polygons

Parallel: A foot divided into twelve inches

A foot divided into tenths and hundredths

both continuing over both halves

In Mer, inclination of the meridian

Lat, the line of latitude

Cho, the line of chords

Hour, the line of hours

shorter scales

Lower side
Sectoral: Tan, the line of natural tangents

Sin, the line of natural sines

Tan, continuation of the line of natural tangents

Parallel: Tan, the line of artificial tangents

T.Sin, the line of artificial tangents and sines for small angles

Sin, the line of artificial sines

Num, Gunter’s line of numbers (logarithms)

The principal changes are: (a) the interchange of the chord and natural sine sectoral scales to a more logical arrangement, (b) the addition of a line of artificial sines/tangents for small angles, and (c) the four short additional scales, primarily for use with sun dials, that have now become standard.

The illustration appears to be of a brass sector as there is no arch shown but, as will become apparent, these are, with perhaps a couple of minor changes, the standard scales used on arch jointed, ivory sectors.

Figure 4 - Gilbert & Sons, London, arch joint sector

Figure 4 shows an arch joint sector from my collection by Gilbert & Sons that can be dated between 1806 and 1819 according to Gloria Clifton.v

The scales on this instrument, which is actually quite a late arch joint sector, are essentially the same as those in Robertson’s illustration. The sectoral lines are the same but the decimal foot and ST scales have been omitted from the parallel lines and the NUM scale has been extended.

Now arch joint sectors were always (as far as I know) signed on the hinge plate on the side that contains the sine and tangent sectoral scales and thus photographs in books always show this side. However, apart from a few early examples, round joint sectors are signed on the other side that contains the chord, secant and polygon sectoral scales. Peter Hopp’s article, except for the first two figures, only ever showed the signed side and thus the side shown for the arch joint sectors was the opposite to that shown for the round joint sectors. This led Peter to conclude that the scale arrangements were significantly different between the two types. However, I doubted this as I knew that the signed side of the arch joint sector had basically the same set of scales as the unsigned side of the round joint sector. The illustration of that side in Peter’s article was too small and indistinct to show that. His reasoning based on the two Dollond sectors illustrated on page 74 was therefore invalid.

My problem was that I did not anywhere have a picture of the unsigned side of an arch joint sector and until recently I had been unable to purchase or examine one in detail. My recent purchase of the Gilbert & Sons example has closed the loop and yes there are differences. Thus on the round joint sector the four dialling scales (Inclination of the Meridian, etc.) seen on the reverse of the arch joint sector have been deleted and the decimal inch scale has been resurrected as an edge scale along the outer edge of the rule. Also the order of the artificial sine and tangent scales has been reversed.

It now becomes clear why the signature on round joint rules moved to the opposite side. With the deletion of the dialling scales there was now more space on that side to place the “maker’s” name and address. Why were the dialling scales deleted? I suspect this was due to a combination of reasons:

·	The scales were available on either a plain scale or protractor in the same set of instruments

·	Watches were becoming more accurate and needed less reference to a sundial

·	There was a move from local time to standard (Greenwich) time, especially once railways became commonplace and the telegraph had been introduced.

I have been unable to determine whether some arch joint sectors had a decimal foot scale along the outer edges but this may have been so.

In the same manner as the evolution from brass to the arch joint ivory type, there was a gradual transition from the arch joint design to the round joint one. This took place at the end of the 18th and start of the 19th centuries. Early in the 19th century the dialling and navigational scales on plain scales and rectangular protractors were also deleted in favour of more fractional scales for some of the same reasons as they had disappeared, probably a little earlier, from sectors.

All the sectors so far discussed have been six-inch examples. I have two 4½-inch sectors in my collection. The scales are the same. There were also twelve-inch sectors made. An illustration in Instruments of the Imagination by Howard Dawes of an ivory arch joint sector by Thomas Heathvi appears to show an additional short pair of sectoral lines on the signed side. It is too indistinct to see any more detail. Whilst studying this picture I noticed that it also includes a six-inch, arch joint sector viewed from the reverse side that does not have the dialling scales - something I had not noticed until I came to write this article.

Figure 5 - T & T Blunt, London, round joint sector

Figure 5 shows a sector by T & T Blunt, which dates from between 1801 & 1822. The scales are as follows:
Signature side:
Sectoral: L, the line of lines

s, the line of natural secants

C, the line of chords

POL, the line of polygons

Parallel: A continuous foot divided into twelve inches

Reverse side:
Sectoral: S, the line of natural sines

t, the continuation of the line of tangents

T, the line of tangents

Parallel: t, the line of artificial tangents

s, the line of artificial sines

n, Gunter’s line of numbers (logarithms)

Outer edge: A foot rule divided into tenths and hundredths

All of my round joint sectors have this scale arrangement. So, who were the actual makers? I had a number of possible theories:

1.	There were very many different makers of the whole sector

2.	There were a few makers of hinged ivory blanks and many finishers from that state

3.	There were a few suppliers of ivory leaves and also suppliers of brass hinges that the instrument makers bought and finished

4.	The dividing of the blanks was done by a few specialist firms with dividing engines

5.	There were only a few actual makers of sectors (Peter’s theory)

6.	Some ‘makers’ bought sectors from other makers who were better equipped to make sectors.

To test these theories might require far more sectors than in any one collection as only a few would come from any particular time period and unsigned ones could not be dated unless they matched a signed one in every detail. To determine that two sectors were actually made by the same maker would require a close or exact match for every detail. These details include:

·	The length, shape and spacing of the hinge plates

·	The riveting

·	The imprint of the letter and number stamps

·	The star patterns

Since the letter and number stamps would quite likely have been made by the instrument maker himself, if there are two relatively early instruments with identical stamp impressions then they must surely have been made by the same maker. To be certain they are identical the whole sector must be examined and the detail must be sharp, a requirement that the web pictures do not meet.

Fortunately I have four sectors that date from about 1810 + a few years:

·	Gilbert & Sons (arch joint)

·	T & T Blunt

·	Jacob & Halse (from a Shagreen case)

·	An unmarked 4½ inch sector in a sharkskin case

Maybe it’s just lucky coincidences but the lettering/numbering on the first two appears a pretty exact match suggesting the same maker for both of these, and the lettering/numbering on the second pair is quite different from the first pair but a pretty exact match again between the two. This suggests that there were at least two makers of sectors in that period who made sectors for other ‘makers’ as well.

To see if any more of my sectors might be compared with one or more others - i.e. be from the same approximate date, I created a table in approximate date order. An * denotes a date range based on the case and the drawing instruments within it.

Signature	Body	Hinge	Size	Date range
W & S Jones	Ivory	Brass	6 inch	1792-1800
Gilbert & Sons	Ivory	Brass, arch	6 inch	1806-19
Jacob & Halse	Ivory	Brass	6 inch	1809-10
T & T Blunt	Ivory	Brass	6 inch	1801-22
Bleuler	Ivory	Brass	6 inch	1790-1822
Unsigned	Ivory	Brass	4.5 inch	1800-30*
A Abraham	Ivory	Brass	6 inch	1800-30*
Unsigned	Boxwood	Brass	6 inch	1800-30*
Williams & Haydon	Ivory	Brass	6 inch	1820-40
Watkins & Hill	Ivory	Brass	6 inch	1822-56
J Archbutt	Ivory	Electrum	6 inch	1838-64
Unsigned	Ivory	Electrum	6 inch	1840-60*
Elliott Bros	Ivory	Brass	6 inch	1854-86
J Casartelli	Ivory	Electrum	6 inch	1852-96
Unsigned	Ivory	Brass	6 inch	19th century
Unsigned	Boxwood	Brass	6 inch	19th century
Unsigned	Boxwood	Brass	4.5 inch	19th century
Unsigned	Boxwood	Brass	6 inch	1860-1900*
Unsigned	Ivory	Electrum	6 inch	1840-1900*
Stanley	Ivory	Electrum	6 inch	1870-1890*
Unsigned	Ivory	Brass	6 inch	1880-1900*

Before continuing to examine and compare individual sectors it is worth observing that sectors ceased to appear in the catalogues of drawing instrument manufacturers by early in the 20th century. This is not surprising as the Mannheim type of slide rule was readily available by then. Indeed Stanley quoted as early as 1878vii (and possibly earlier?) “The SECTOR is a kind of twofold rule, commonly supplied with a case of mathematical instruments, as a kind of established ornament, in most instances to be practically used only as a kind of bevel to erect angles…….”. None the less Stanley continued to supply them in his sets as long as there was a demand and they appeared as one line in the price list bound into the 1900 edition of his book. They do not, however appear in the 1912 Stanley catalogue and I think it is fair to assume that virtually all sectors are pre-1900.

Examination of the table shows that there are three other sectors from the same approximate time frame as the four already compared. The Abraham one is a pretty good match for the Jacob & Halse one, probably close enough to conclude it is a third by that maker. Holding both these sectors to the light, and ivory being translucent, appears to clinch it. Both sets of metal leaves are actually arched with exactly the same profile inside the ivory - the hinges are unique in shape and identical. This is also true of the 4½ inch ivory sector. The Bleuler sector is also a pretty good match for the Jacob & Halse one.

The other sector examined was an unsigned boxwood one from a sharkskin case. The hinge circular part was of smaller radius than any of the ivory ones in this group but the letters and numbers bore a fair resemblance to those of the Gilbert and Blunt sectors. This alone was not enough to conclude it was made by the same maker as those two. However there is a further feature that does match. Near the inner end of the polygon scales, instead of very small numbers being used as on other sectors, the numbers 10, 11 and 12 are much larger and stamped across the edges of the two leaves. This is illustrated in Figure 6. So we again probably have a third sector by the same maker; i.e. we have seven sectors from this period but probably only two makers.

Figure 6 - Detail of the T & T Blunt sector showing large 10,11 & 12

However, all is not perhaps as simple as that. I next looked at the slightly earlier W & S Jones one. Holding it to the light the hinge leaves are again arched although the profile and length of the hinge plates is slightly different. The lettering and numbering is very similar to that on the T & T Blunt sector and once again the 10, 11 and 12 are much larger and stamped across the edges of both leaves. Holding the Blunt sector to the light again reveals the arch and the hinge turns out to be identical to those of the Jacob & Halse trio.

So, I now have to revise my conclusion; there is one hinge maker supplying hinges to two different sector makers, each of whom make sectors for at least three firms.

The slight differences in the numbers and letters between the Jones and Blunt sectors can be easily explained. The stamps would wear with use and were probably re-made several times within the time period.

Let’s return to Bleuler. This was the most recently obtained ivory sector, obtained after I had prepared the first draft of this article. There is a picture of another Bleuler sector on page 333 of the Rule Bookviii but this clearly belongs to the Gilbert and Blunt group. Thus we have two sectors from one maker, one of each of the two basic types for that period clearly indicating that they were bought in and not made by Bleuler.

The next sectors in the table are those by Williams and Haydon, and by Watkins and Hill. It is immediately clear that these are not both by the same maker. There are significant differences between the lettering styles. Holding to the light reveals that the Williams and Haydon one has the internally arched hinge whilst the Watkins and Hill one is cut straight across inside. Both are brass. The Williams and Haydon one again has the large 10, 11 and 12 numbers on the polygon scales and the similarity of the lettering and numbering suggests a later instrument by the same maker as the Blunt and Gilbert sectors. The Watkins and Hill one is possibly later than the Williams and Haydon one and the style of its lettering and numbering is similar to that of the Abraham, Jacob & Halse group so it could be a later product of the same firm. The two distinct styles of lettering are shown in Figures 7a and 7b.

Figure 7 - The two distinctly different styles of scale lettering. The Gilbert & Sons on the left with the large L and C is typical of the Gilbert/Blunt group and the Watkins & Hill one on the right is typical of the Jacob & Halse/Abraham group. Note also the different dividing of the L, line of lines, scales.

Next in sequence is the J Archbutt sector. The hinge leaves now exhibit a double arch inside and, whilst the lettering/numbering styles have a resemblance to those in the Abraham, Jacob & Halse Group they are not close enough to conclude the same maker. The link has been broken. There are too few signed sectors from the later part of the 19th century to be able to draw further conclusions about makers. However yet another sector has the large 10, 11 & 12 on the polygons scales bridging the gap between the two leaves although the style of the other lettering and numbering does not match the earlier sectors with this feature. It has an electrum hinge with a slight double arch and dates from 1840-60.

The Casartelli sector again has internal double arch electrum hinge leaves but the lettering and numbering does not closely match others and the sector is actually wider than the earlier ones. The Elliott Bros one may be earlier as it has a brass, single internal arch, hinge. The lettering and numbering style is again similar to that of the Abraham, Jacob and Halse group but it has the large 10, 11 and 12 of the other group so no conclusion can be drawn. It is the same width as the Casartelli one.

The most recent named sector is the Stanley one. The large 10, 11 and 12 on the polygon scales is again present and the hinge is an electrum double inside arch type but the numbers and letters are not a match for any of the other sectors although similar in style. The width is again the same as the Casartelli one.

At some stage in the 19th century off-the-shelf letter and number stamps would have become available (I don’t know when) and this would complicate identifying sectors from the same maker and differentiating between makers.

My tentative conclusions regarding makers therefore are:

·	There were two distinctly different styles of lettering and numbering that persisted through the entire period of manufacture of round joint sectors from at least ca.1790 to ca.1900

·	There were only a few actual makers of ivory and boxwood sectors

·	The hinges were probably made by a specialist maker who supplied them to the sector makers, or the sector makers were supplied by a specialist maker with hinged ivory or boxwood blanks

·	The design of the internal hinge leaves changed from single arch to shallow double arch (i.e. a separate arch in each leaf) in the middle of the 19th century and possibly coincided with the change from brass to electrum on the better quality instruments

·	There is still no evidence to suggest who the actual makers were

·	There is little known about the actual manufacturing methods used, there are no known records of these processes, and more research is needed.

Finally Dr. Alison Morrison-Low has studied the instrument making trade in detail and described this in Making Scientific Instruments in the Industrial Revolution.ix On page 149 she refers to a statement by the prestigious instrument maker George Adams (the elder) that “implies his business was underpinned by a complex trade structure - hidden to historians, yet probably taken for granted by contemporaries - of outworkers and subcontractors, channelling specialist instrument parts to be assembled for sale or finished in Adams’ Fleet Street premises.” Dr Morrison-Low observes elsewhere that mechanisation was slow to be adopted in the instrument trade, much hand work continuing into the middle of the 19th century, and this included the adoption of dividing engines. It should be noted that Ramsden’s early dividing engines were for use with circles and arcs, not linear items such as rules. Also logarithmic and trig scales require significantly more complex dividing engines.

i So who actually made them? Some 19th century slide rule makers or retailers, Peter M Hopp C Eng, MBCS, Slide Rule Gazette Issue 12 Autumn 2011
ii The construction and principal uses of mathematical instruments translated from the French of M. Bion chief instrument maker to the French king by Edmund Stone……..first published in 1758 and reprinted by the Astragal Press in 1995. ISBN 1-879335-60-3
iii Scientific Instruments, Harriet Wynter and Anthony Turner, Studio Vista, 1975, ISBN 0 289 70403 0, Fig 70 P62
iv A treatise of such mathematical instruments as are usually put into a portable case………The third edition with many additions. First published in 1775 and reprinted by Flower-de-Luce Books in 2002, ISBN 1-931468-11-7
v Directory of British Scientific Instrument Makers 1550-1851, Gloria Clifton, Zwemmer, 1995, ISBN 0 302 00634 6
vi Instruments of the Imagination, Howard Dawes, Fig.72, The Dawes Trust Ltd, 2009, ISBN 095510891-8
vii A descriptive treatise on mathematical instruments: their construction, uses, qualities, selection, preservation and suggestions for improvement (1878) William Ford Robinson Stanley, Kessinger Publishing reprint ISBN 0558826528
viii The Rule Book, Measuring for the Trades, Jane Rees and Mark Rees, Astragal Press, 2010, ISBN 978-1-931626-26-2
ix Making Scientific Instruments in the Industrial Revolution, A D Morrison-Low, Ashgate Publishing, 2007, ISBN978-0-7546-5758-3