The second method, commonly known as “back-calculation”, depends on the assumption that, throughout the life of the fish, a constant relationship is maintained between size of fish and size of scale. Since an annulus represents a former scale margin, the size of the fish when this annulus was formed may be computed if this relationship is known. The simplest case is described by the equation:

l = L d / D      (1)

where l is the length of the fish when the annulus was formed, L is the present length, d is the diameter of the annulus, and D is the present diameter of the scale. The measure of fish length employed in the present work was the “fork length” (i.e., the distance from the tip of the premaxilla to the centre of the V-shaped notch in the caudal fin), while the diameter of scale or annulus was the maximum diameter (i e, the width from left to right in Plate 28, Figure 1). Various other linear dimensions either of fish or scale would serve as well, but the above are the most conveniently measured. Since the three terms L, d and D may be readily determined, it is a simple matter to estimate the entire growth record of any fish by computing l for each successive annulus. One of the principal advantages of this method is that the maximum amount of information can be obtained from the minimum number of scale samples.

If any appreciable degree of allometry exists between scale and fish dimensions, equation (1) must be suitably modified, the usual process being to compute from a sample of suitable size the regression of fish length on scale diameter. Examination of a small sample of snapper immediately showed that the relationship was unlikely to be a simple proportional one. For a sample of 25 snapper ranging in size from 6.

Text-fig 1—Back-calculated lengths for age groups 1 to 10 plotted against actual length of fish, for 80 snapper from Hauraki Gulf.

Text-fig 2.—Derivation from Text-fig. 1 of an estimate of the regression curve (A-B-C) of scale diameter on fish length.

to 29 inches the linear regression equation of mean diameter (measured in arbitrary units) of six scales on length of fish was found to be:

Scale diameter = 0.5204 fish length + 0.7480 (2)

With a standard error of 0.1160 the intercept, 0.7480, differs from zero at the 0.1 per cent, level of confidence, so the simple proportional relationship obviously does not hold. As a test of linearity the allometric regression equation was computed:

Scale diameter = 0.7382 fish length ^0.9062 (3)

The exponent, 0.9062, with a standard error of 0.0298, differs from unity at the 1 per cent, level of confidence. Thus it is unlikely that there is a linear relationship between fish length and scale diameter. The exact relationship is less easily determined since, even with the most careful standardisation of the position from which they were taken, scale sizes for any one fish were extremely variable, with a coefficient of variation up to 20 per cent. in some cases.

However, it is not the absolute size of annuli which is significant, but their size relative to that of the scale. Thus it has been possible to reconstruct the approximate shape of the regression curve by a method which is independent of variations in scale diameter. Text-fig. 1 presents the results of scale readings from a sample of 80 snapper ranging in age from one to ten years Lengths back-calculated for each age group by equation 1 are plotted against length of fish from which the scale was taken. Apart from the first age group, there is a distinct tendency for back-calculated lengths to increase with the actual length of the fish. In Text-fig. 2 a series of regression lines have been fitted by eye to the points for each age class. The line O-C, representing equation 1, intersects these regression lines at points which, when projected onto the abscissa, give an estimate of the mean fish length when each succes-

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sive annulus was formed—i.e., the growth rate. (For instance, a fish forming its third annulus will be expected to be about 7.8 inches long.) The ordinate of the graph may now be used as a measure of “scale diameter”; not the diameter of any actual scale, but that of an hypothetical “average” scale with a size variation dependant only upon length of fish. Since only an arbitrary measure is required, the same units may be employed as before, so that the basic unit is 1/14 of the scale diameter of a 14-inch fish. Co-ordinates (broken lines) are now drawn from either end of each regression line to intersect at a series of points which are joined by the curve A-B-C. This is an estimate of the regression of scale diameter on fish length in the range 4–14 inches. Since it was desired to obtain only an approximate picture of the shape of the curve, the diagram has been simplified by making the age group regression lines parallel (with the exception of 1). Thus B-C becomes a straight line with a positive intercept (cf. equation 2) although it is not unlikely that the entire curve A-B-C is curvilinear (cf. equation 3). It is clear then that the simple back-calculation relationship given in equation 1 will give a biased estimate of the growth rate. For instance (following the dotted co-ordinates in Text-fig. 2) the third annulus of a 14-inch fish will have a diameter of 8–9 units, corresponding to a back-calculated length of 8–9 inches, as compared with the true value, 7.8 inches. Since, in these investigations, there was virtually no limit to the number of scale samples which could be obtained, the back-calculation method was abandoned in favour of the simpler method by which each fish yields only its length and age at time of capture.

Three different techniques were employed in the reading of scales. In the first stages of the investigation they were simply placed dry under a low-power stereoscopic microscope. Later a reading apparatus was constructed in which the scale was placed in a small projector which was directed vertically downward onto a mirror, so inclined that the image was projected onto an inclined ground-glass screen immediately in front of the projection stage. The whole apparatus was enclosed in a plywood case and recessed into a bench so that the screen was at a convenient working level. By this means the magnified image could be viewed with a minimum of eye strain in ordinary room lighting and, if necessary, measurements could be taken with a celluloid rule. The scale was usually examined dry, being held flat between two glass plates, but provision was made for water or glycerine mounts in the few cases where these were considered necessary. Usually little preparation was needed beyond washing in tap-water and rubbing off any foreign matter between thumb and forefinger. In the third method the scale was placed on a matt black surface and viewed with the unaided eye by oblique incident illumination. By this means some determinations could be made very easily and rapidly, but in cases of uncertainty the projector was always employed.

In every case at least five scales were examined, regenerated scales (which do not have a complete growth record) being rejected. If the first five gave inconclusive results a second five were examined, but it was found that little could be gained by increasing the number above ten. Results were recorded on a cyclo-styled sheet with appropriate spaces for: serial number, length of fish (inches and tenths), sex, number of annuli, alternative annulus count (if any), reliability of reading, number of scales examined, and remarks. In the reliability column was entered a code letter ranging from “A” where the reading was clear and consistent in all scales, to “D” where no interpretation was possible. Though this classification is largely subjective, it was of considerable value in comparing the interpretations of different observers. It was found, for instance, that if both placed a set of scales in category “A” there was seldom if ever any disagreement over the number of annuli counted. Determinations not classed as “A” were checked by a second observer, while in summarizing results “C” and “D” categories were rejected as unreadable. As a check on personal errors 100 sets of scales of snapper ranging from 3 to 6 inches in length were read independently by three different persons. It was found that in only 53