scale (n.1) Look up scale at Dictionary.com
"skin plates on fish or snakes," c.1300, from Old French escale "cup, scale, shell pod, husk" (12c., Modern French écale) "scale, husk," from Frankish *skala or some other Germanic source, from Proto-Germanic *skælo "split, divide" (cognates: Dutch schaal "a scale, husk," Old High German scala "shell," Gothic skalja "tile," Old English scealu "shell, husk"), from PIE root *(s)kel- (1) "to cut, cleave, split" (cognates: Latin culter "knife," scalpere "to cut, scrape;" Old Church Slavonic skolika "mussel, shell," Russian skala "rind, bark," Lithuanian skelti "split," Old English scell "shell," scalu "drinking cup, bowl, scale of a balance").

In reference to humans, as a condition of certain skin diseases, it is attested from c.1400. As what falls from one's eye when blindness ends (usually figurative), it echoes Acts ix:18 (Latin tanquam squamæ, Greek hosei lepides).
scale (n.2) Look up scale at Dictionary.com
weighing instrument, early 15c.; earlier "pan of a balance" (late 14c.); earlier still "drinking cup" (c.1200), from Old Norse skal "bowl, drinking cup," in plural, "weighing scale" from a noun derivative of Proto-Germanic *skæla "split, divide" (cognates: Old Norse skel "shell," Old English scealu, Old Saxon skala "a bowl (to drink from)," Old High German scala, German Schale "a bowl, dish, cup," Middle Dutch scale, Dutch schaal "drinking cup, bowl, shell, scale of a balance"), from PIE root *skel- (1) "to cut" (see scale (n.1)).

The connecting sense seems to be of half of a bivalve ("split") shell used as a drinking cup or a pan for weighing. But according to Paulus Diaconus the "drinking cup" sense originated from a supposed custom of making goblets from skulls (see skull). Related: Scales. This, as a name for the zodiac constellation Libra, is attested in English from 1630s.
scale (v.1) Look up scale at Dictionary.com
"to climb by or as by a ladder," late 14c., from scale (n.) "a ladder," from Latin scala "ladder, flight of stairs," from *scansla, from stem of scandere "to climb" (see scan (v.)). Related: Scaled; scaling.
scale (n.3) Look up scale at Dictionary.com
"series of registering marks to measure by; marks laid down to determine distance along a line," late 14c., from Latin scala "ladder, staircase" (see scale (v.1)). Meaning "succession or series of steps" is from c.1600; that of "standard for estimation" (large scale, small scale, etc.) is from 1620s. Musical sense (1590s), and the meaning "proportion of a representation to the actual object" (1660s) are via Italian scala, from Latin scala.
scale (v.3) Look up scale at Dictionary.com
"weigh in scales," 1690s, from scale (n.2). Earlier "to compare, estimate" (c.1600). Meaning "measure or regulate by a scale" is from 1798, from scale (n.3); that of "weigh out in proper quantities" is from 1841. Scale down "reduce proportionately" is attested from 1887. Scale factor is from 1948. Related: Scaled; scaling.
scale (v.2) Look up scale at Dictionary.com
"remove the scales of (a fish, etc.)," c.1400, from scale (n.1). Intransitive sense "to come off in scales" is from 1520s. Related: Scaled; scaling.