A method for unbiased capillary number estimation based on the estimation of the Euler-Poincaré characteristic or Euler number with the disector is evaluated. The generation of a new capillary creates a new loop in the capillary network corresponding to a change of exactly one unit in the Euler number of the network. In this way the Euler number has a one-to-one relationship to the number of capillaries in a network. In this report rat glomerular capillaries are counted. Three tissue blocks from each of 14 perfusion-fixed rat kidneys, aged between 5 days and 18 months, were uniformly sampled and embedded in Epon. Two complete glomerular profiles were sampled per block. Three consecutive sections were studied per glomerulus using the middle one to evaluate the topological events of the capillaries in the sampled glomerulus. The use of complete glomerular profiles eliminated problems with the edges, whereas the disector ensured unbiased sampling in the third dimension. Estimates per animal were weighted averages over blocks, the weights being half the number of glomerular profiles in a section from a block. The number of capillaries in a glomerulus is the mean Euler number per disector volume multiplied by the mean volume of glomeruli obtained by the fractionator. The observed coefficient of variation between animals was 18.3% for the estimated number of capillaries per glomerulus, and the observed coefficient of error at the level of blocks within animals was 14.7%. The exact capillary number from the node-branch network of two reconstructed glomeruli equalled the capillary number obtained by the total Euler number from the same glomeruli. This shows in an applied example, as would be expected, that the estimation of capillary number using the Euler number is unbiased and independent of the direction of sectioning.