NavList:

Frank, it seems to me there are different ways to understand your question:

1. Why did sextant manufacturers offer the option of a pentaprism? - At the most basic: to extend the measuring range by 90°, beyond the usual 120° (or so) of the standard sextant. Obviously, for astronomical navigation, there rarely is a need for a range in excess of 90°, while for surveying applications, a range extending to 180° or so often will be useful. Huibert-Jan Lekkerkerk, who has a special interest in collecting surveying sextants, writes that, while the use of the sextant for surveying is documented from the 18th century, specialised surveying sextants appeared in the second half of the 19th century, most of which were technically quintants measuring up to 140° (https://hydrography.pro/presta17/instrumentblog/the-survey-sextant-outside-the-netherlands.html). Accoridng to Huibert-Jan, pentaprisms were an addition to "many later survey sextants". In his online collection, he shows sextants with a pentaprism, or with a provision for attaching a pentaprism, from de Koningh, Observator, C. Plath, and Weems and Plath (https://hydrography.pro/presta17/25-sextants?q=Type+of+use-Survey).The earliest of these is a de Koningh  from 1935. There is also a Husun sounding sextant with an unusual arrangment in which the pentaprism is attached in front of the telescope instead of behind the horizon mirror (https://hydrography.pro/presta17/quintant/297-husun-sounding-sextant.html). With the exception of this Husun model, all other sextants with a pentaprism option in Huibert-Jan's collection, as well as the Freiberger and Cassens & Plath ones, have a standard measuring range up to 125°. Attaching a pentaprism allowed the sextant makers to extend the range of their standard models far beyond 125° without modifying the construction of the sextant, which probably would have been a less desirable solution from a cost-benefit perspective. Also, without a prism, taking sights at greater angles becomes more difficult, as the more the index mirror is inclined, the narrower its reflection in the horizon mirror, and consequently, the larger the area of the horizon mirror that reflects what is behind the observer, rather than what is reflected by the horizon mirror. For astronomical observations, the impact of this effect is alleviated by the fact that the observed body, the horizon, and whatever is behind the observer are usually visually quite distinct and can therefore be distinguished from each other without too much effort. However, with terrestrial observations, where the objects seen on the horizontal plane can be very similar too each other, often providing little contrast, distinguishing the reflected image of the index mirror from the reflection of what is behind the observer and from what is seen through the horizon mirror can become a challange. With a pentaprism, the index mirror is only slightly inclined when measuring angles around 90°, and thus its reflection will cover most of the surface area of the horizon mirror, thus greatly simplifying sights at greater angles.

2. Why use a pentaprism instead of a pentamirror? - In principle, the same benefits could be obtained by using a pentamirror arrangement, that is, two mirrors fixed in the same position as the mirroring surfaces of the pentaprism. In photography, the advantages of using a pentaprism vs. a pentamirror in a SLR/DSLR camera have been discussed extensively (e. g., https://photographyinsider.info/pentaprism-vs-pentamirror-which-is-better/). A pentamirror is lighter and cheaper, but the two mirrors are more difficult to keep clean (which might have been a consideration for surveying sextants used at sea). Also, the two mirrors must be aligned with a high degree of precision, and th alignment must be maintained under thermal and mechanical stress, while the precision of a pentaprism is determined at the point of manufacture and will remain the same until the prism is destroyed. On the one certificate I have seen for a Freiberger pentaprism, a deviation of only 12 arcseconds from a perfect 90 degrees was stated. It might be difficult to achieve and maintain the same degree of precision with a pentamirror arrangement. Furthermore, if the two mirrors of the pentamirror arrangement are coated on the backside (second surface), the light will have two enter and exit a glass medium twice, while only once in a pentaprism, resulting in better optical performance of the latter. This last disadvantage presumaby could be avoided by using first-surface mirrors for the pentamirror; however, before the late 1970s/1980s, first-surface mirrors generally seem not to have been used for sextants (certainly not for C. Plath, Cassens & Plath, or Freiberger sextants), implying that the light would have had to enter and exit four glass surfaces in total (horizon mirror, index mirror, two pentamirror surfaces), potentially impacting optical quality. Also, pentaprisms were widely used (and presumably still are) in terrestrial geodetic surveying. Therefore, they would have been an obvious choice for surveying sextants. This would have been particularly true for Freiberger, a company that has been making geodetic instruments for 250 years and started to produce sextants only by historical accident 70 years ago. All in all, a pentaprism seems like an obvious solution for a mid-20th-century hydrographer or other professional who needed to measure angles of 90° or higher and required an instrument that could be relied upon to provide reasonable accuracy at sea in sometimes rough conditions.

3. Why care about sextant pentaprisms today? -  I suppose the answer will be same as for the question why we think that knowledge about the use of sextants is worth preserving. For demonstrating the concept, an arrangement as you describe is certainly all that is needed. However, the fact that Cassens & Plath (and presumably Freiberger) continue to sell sextant pentaprisms indicates that there are still some customers who feel that for their use case, 1200 Euro is a worthwhile investment.

Martin