An investigation at sea level of cosmic-ray showers with sizes from 5×105 to over 109 particles is described. The core locations, arrival directions, and particle density distributions of several thousand showers whose cores landed within an area of 105 m2 were determined by the techniques of fast-timing and density sampling. The most important results are as follows: (1) The existence of primary particles with energies greater than 1018 ev is established by the observation of one shower with more than 109 particles. (2) The function f(r)=0.45(NR02)r-0.7(1+r)- 3.2, where r=RR0 and R0=79 m, describes the lateral distribution of particles at distances in the range 50 m<R<400 m and for showers with sizes in the range 5×105<N<108. (3) At distances greater than 50 m from the core the density fluctuations in individual showers have a Poisson distribution. (4) The size and zenith angle distribution can be represented by the formula s(N, x)=s0(106N)Γ+1exp[-(x- x0)Λ], where x=x0θ, x0=1040 g cm-2, s0=(6.6+/-1.0)×10-8 cm-2 sec-1 sterad-1, Γ=1.9+/-0.1, Λ=(113+/-9) g cm-2, x0<x<1.3x0, and 7×105<N<7×108. (5) No evidence is found of anisotropy in the arrival directions or of a break in the energy spectrum of the primaries up to the largest energies observed. (6) Assuming a specific model for shower development and taking into account fluctuations in the depth of the first interaction, the integral energy spectrum of the primaries is J(E)=J0(1015E)γ, where J0=(8.1+/-3.1)×10-11 cm-2 sec-1 sterad-1, γ=2.17+/-0.1, and 3×1015 ev<E<1018 ev.