Odd, but I found this same Smithsonian sextant on the following site, also related to the NMAH:http://americanhistory.si.edu/collections/search/object/nmah_1087501
The description includes the serial number 5670 (rather than 5760 found on the other site) along with Navy 4705, but this description also mentions that it is in box number 3946, and also includes a photo of the sextant in the box where the number is visible. see below.
This sextant was probably made for use in World War I. Brandis termed it a "U.S. Navy surveying sextant, 6 inch radius, reading 30 seconds of arc." New it cost $120. The frame is anodized brass. The silvered scale is graduated every 10 minutes from -5° to +182° and read by vernier with tangent screw and magnifier. The inscriptions read "BRANDIS & SONS, BROOKLYN, N.Y." and "5670." The words "U.S. Navy-N-4705" are scratched onto the arc. The serial number inside the wooden box is 3946.
Ref: Brandis & Sons Mfg. Co., Catalogue No. 20. Instruments of Precision (Brooklyn, n.d.), p. 298.
Unfortunately, the photo is not of high enough resolution to see the serial number, so it is hard to tell whether the actual serial is 5670 or 5760. My guess is that 5670 is a typo.
Can anyone find a copy of this publication?
Brandis & Sons Mfg. Co., Catalogue No. 20. Instruments of Precision (Brooklyn, n.d.)